Advanced probability and applications
COM-417
Media
16, COM-417: Lecture 5.3
25.05.2020, 08:57
Joint distribution of a Gaussian random vector
11, COM-417: Lecture 4.1
25.05.2020, 08:52
Expectation of a random variable
23, COM-417: Lecture 7.3
25.05.2020, 09:12
Kolmogorov's 0-1 law
4, COM-417: Lecture 1.3
25.05.2020, 08:38
Sigma-field generated by a collection of random variables
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
APA_lecture14b_partI
01.06.2023, 15:27
APA_lecture11b_partI
11.05.2023, 10:33
22, COM-417: Lecture 7.2 bis
25.05.2020, 09:09
A remark about the proof of the LLN
6, COM-417: Lecture 2.2
25.05.2020, 08:40
Distribution of a random variable
27, COM-417: Lecture 9.1
25.05.2020, 09:16
The central limit theorem
34, COM-417: Lecture 10.3
25.05.2020, 09:31
Concentration: Hoeffding's inequality
23, COM-417: Lecture 7.3
25.05.2020, 09:12
Kolmogorov's 0-1 law
18, COM-417: Lecture 6.2
25.05.2020, 09:02
Various types of convergence of sequences of random variables
APA_lecture14a_partI
31.05.2023, 16:00
APA_lecture12a_partI
17.05.2023, 16:22
10, COM-417: Lecture 3.3
25.05.2020, 08:50
Convolution
APA_lecture10b_partI
04.05.2023, 10:37
7, COM-417: Lecture 2.3
25.05.2020, 08:41
The Devil's staircase
45, COM-417: Lecture 13.2c
25.05.2020, 09:43
Proof of the martingale convergence theorem (part 3)
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
COM-417 Advanced Probability and Applications
15.04.2023, 13:27
6, COM-417: Lecture 2.2
25.05.2020, 08:40
Distribution of a random variable
30, COM-417: Lecture 9.3
25.05.2020, 09:23
Proof of the CLT using characteristic functions
34, COM-417: Lecture 10.3
25.05.2020, 09:31
Concentration: Hoeffding's inequality
APA_lecture13a_partI
24.05.2023, 16:21
APA_lecture13b_partII
25.05.2023, 10:00
30, COM-417: Lecture 9.3
25.05.2020, 09:23
Proof of the CLT using characteristic functions
40, COM-417: Lecture 12.2
25.05.2020, 09:38
Sub- and super-martingales, stopping times
2, COM-417: Lecture 1.1
25.05.2020, 08:35
Sigma-fields, sigma-field generated by a collection of events
Here is a small extra comment regarding the definition of a sigma-field:
From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n
13, COM-417: Lecture 4.3
25.05.2020, 08:53
Characteristic function
31, COM-417: Lecture 10.1a
25.05.2020, 09:27
The coupon collector problem (part 1)
APA_lecture12a_partII
17.05.2023, 16:23
APA_lecture14b_partII
01.06.2023, 15:27
22, COM-417: Lecture 7.2 bis
25.05.2020, 09:09
A remark about the proof of the LLN
23, COM-417: Lecture 7.3
25.05.2020, 09:12
Kolmogorov's 0-1 law
8, COM-417: Lecture 3.1
25.05.2020, 08:43
Independence of two events, random variables, sigma-fields
One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:
- To prove that two random variables are not independent, it suffices indeed to find one counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).
- On the contrary, independence requires that equality holds for all (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
39, COM-417: Lecture 12.1
25.05.2020, 09:37
Martingales
11, COM-417: Lecture 4.1
25.05.2020, 08:52
Expectation of a random variable
APA_lecture13b_partI
25.05.2023, 09:59
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
16, COM-417: Lecture 5.3
25.05.2020, 08:57
Joint distribution of a Gaussian random vector
6, COM-417: Lecture 2.2
25.05.2020, 08:40
Distribution of a random variable
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
APA_lecture14a_partII
31.05.2023, 16:00
7, COM-417: Lecture 2.3
25.05.2020, 08:41
The Devil's staircase
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
41, COM-417: Lecture 12.3
25.05.2020, 09:39
The optional stopping theorem, with a proof
26, COM-417: Lecture 8.3
25.05.2020, 09:15
The Curie-Weiss model
34, COM-417: Lecture 10.3
25.05.2020, 09:31
Concentration: Hoeffding's inequality
18, COM-417: Lecture 6.2
25.05.2020, 09:02
Various types of convergence of sequences of random variables
9, COM-417: Lecture 3.2
25.05.2020, 08:44
Independence of more sigma-fields
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
APA_lecture13a_partI
24.05.2023, 16:21
16, COM-417: Lecture 5.3
25.05.2020, 08:57
Joint distribution of a Gaussian random vector
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
APA_lecture13b_partII
25.05.2023, 10:00
30, COM-417: Lecture 9.3
25.05.2020, 09:23
Proof of the CLT using characteristic functions
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
APA_lecture10b_partII
04.05.2023, 10:37
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
3, COM-417: Lecture 1.2
25.05.2020, 08:37
Sub-sigma-fields, random variables
4, COM-417: Lecture 1.3
25.05.2020, 08:38
Sigma-field generated by a collection of random variables
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
APA_lecture10b_partI
04.05.2023, 10:37
APA_lecture12a_partI
17.05.2023, 16:22
40, COM-417: Lecture 12.2
25.05.2020, 09:38
Sub- and super-martingales, stopping times
43, COM-417: Lecture 13.2a
25.05.2020, 09:42
Proof of the martingale convergence theorem (part 1)
35, COM-417: Lecture 11.1
25.05.2020, 09:32
Concentration: large deviations principle
APA_lecture13b_partII
25.05.2023, 10:00
APA_lecture11b_partI
11.05.2023, 10:33
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
COM-417 Advanced Probability and Applications
APA_lecture14b_partIII
01.06.2023, 15:27
APA_lecture14b_partII
01.06.2023, 15:27
APA_lecture14b_partI
01.06.2023, 15:27
APA_lecture14a_partII
31.05.2023, 16:00
APA_lecture14a_partI
31.05.2023, 16:00
APA_lecture13b_partII
25.05.2023, 10:00
APA_lecture13b_partI
25.05.2023, 09:59
APA_lecture13a_partII
24.05.2023, 16:21
APA_lecture13a_partI
24.05.2023, 16:21
APA_lecture12a_partII
17.05.2023, 16:23
APA_lecture12a_partI
17.05.2023, 16:22
APA_lecture11b_partII
11.05.2023, 10:34
APA_lecture11b_partI
11.05.2023, 10:33
APA_lecture10b_partII
04.05.2023, 10:37
APA_lecture10b_partI
04.05.2023, 10:37
APA_lecture9b_partII
27.04.2023, 10:23
APA_lecture9b_partI
27.04.2023, 10:22
APA_lecture8b_partII
20.04.2023, 10:39
APA_lecture8b_partI
20.04.2023, 10:38
APA_lecture8a_partII
19.04.2023, 16:27
APA_lecture8a_partI
19.04.2023, 16:20
53, COM-417 Complement: Convergence in distribution
30.04.2021, 11:33
An interesting counter-example
52, COM-417 Complement: Monotone class theorem
14.04.2021, 19:03
51, COM-417 Complement: Some examples of cdfs
03.03.2021, 20:22
50, COM-417: Lecture 14.2b
25.01.2021, 10:07
McMiarmid's inequality
49, COM-417: Lecture 14.2a
25.01.2021, 10:06
Azuma's inequality
48, COM-417: Lecture 14.1b
24.01.2021, 22:05
Generalization of the MCT to sub- and supermartingales (part b)
47, COM-417: Lecture 14.1a
24.01.2021, 22:04
Generalization of the MCT to sub- and supermartingales (part a)
54, COM-417 Complement: Extra lecture on R0
25.05.2020, 10:00
Extra lecture about branching processes, recorded in April 2020
After recording this, I realized it would perhaps be more appropriate to refer to "R" in this lecture, rather than to "R0": R0 is indeed the "basic reproduction number" of the virus, estimated around 3.5, on which one has no control; while R is its "effective reproduction number", on which our behaviour can have a great influence.
46, COM-417: Lecture 13.3
25.05.2020, 09:44
The martingale convergence theorem (version 2)
45, COM-417: Lecture 13.2c
25.05.2020, 09:43
Proof of the martingale convergence theorem (part 3)
44, COM-417: Lecture 13.2b
25.05.2020, 09:43
Proof of the martingale convergence theorem (part 2)
43, COM-417: Lecture 13.2a
25.05.2020, 09:42
Proof of the martingale convergence theorem (part 1)
42, COM-417: Lecture 13.1
25.05.2020, 09:40
The martingale convergence theorem (version 1)
Note: contrary to what is said at the beginning of the video, this is not the last lecture of the semester: there is still one to come (Lecture 14).
41, COM-417: Lecture 12.3
25.05.2020, 09:39
The optional stopping theorem, with a proof
40, COM-417: Lecture 12.2
25.05.2020, 09:38
Sub- and super-martingales, stopping times
39, COM-417: Lecture 12.1
25.05.2020, 09:37
Martingales
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
35, COM-417: Lecture 11.1
25.05.2020, 09:32
Concentration: large deviations principle
34, COM-417: Lecture 10.3
25.05.2020, 09:31
Concentration: Hoeffding's inequality
33, COM-417: Lecture 10.2
25.05.2020, 09:30
Moments and Carleman's theorem
Here is a link to the Online Encyclopedia of Integer Sequences mentioned at the end of the above video: oeis.org (@Neil Sloane)
32, COM-417: Lecture 10.1b
25.05.2020, 09:28
The coupon collector problem (part 2)
31, COM-417: Lecture 10.1a
25.05.2020, 09:27
The coupon collector problem (part 1)
30, COM-417: Lecture 9.3
25.05.2020, 09:23
Proof of the CLT using characteristic functions
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
27, COM-417: Lecture 9.1
25.05.2020, 09:16
The central limit theorem
26, COM-417: Lecture 8.3
25.05.2020, 09:15
The Curie-Weiss model
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
24, COM-417: Lecture 8.1
25.05.2020, 09:13
St Petersburg's paradox and extension of the weak law
Sorry, as the beginning of this one is a bit chaotic; also, it is slightly longer than the others... but it is worth the effort, I think!
23, COM-417: Lecture 7.3
25.05.2020, 09:12
Kolmogorov's 0-1 law
22, COM-417: Lecture 7.2 bis
25.05.2020, 09:09
A remark about the proof of the LLN
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
19, COM-417: Lecture 6.3
25.05.2020, 09:03
Almost sure convergence implies convergence in probability
18, COM-417: Lecture 6.2
25.05.2020, 09:02
Various types of convergence of sequences of random variables
17, COM-417: Lecture 6.1
25.05.2020, 09:00
Cauchy-Schwarz, Jensen and Chebyshev inequalities
16, COM-417: Lecture 5.3
25.05.2020, 08:57
Joint distribution of a Gaussian random vector
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
13, COM-417: Lecture 4.3
25.05.2020, 08:53
Characteristic function
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
11, COM-417: Lecture 4.1
25.05.2020, 08:52
Expectation of a random variable
10, COM-417: Lecture 3.3
25.05.2020, 08:50
Convolution
9, COM-417: Lecture 3.2
25.05.2020, 08:44
Independence of more sigma-fields
8, COM-417: Lecture 3.1
25.05.2020, 08:43
Independence of two events, random variables, sigma-fields
One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:
- To prove that two random variables are not independent, it suffices indeed to find one counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).
- On the contrary, independence requires that equality holds for all (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).
7, COM-417: Lecture 2.3
25.05.2020, 08:41
The Devil's staircase
6, COM-417: Lecture 2.2
25.05.2020, 08:40
Distribution of a random variable
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
4, COM-417: Lecture 1.3
25.05.2020, 08:38
Sigma-field generated by a collection of random variables
3, COM-417: Lecture 1.2
25.05.2020, 08:37
Sub-sigma-fields, random variables
1, COM-417: Introduction and notations
25.05.2020, 08:36
2, COM-417: Lecture 1.1
25.05.2020, 08:35
Sigma-fields, sigma-field generated by a collection of events
Here is a small extra comment regarding the definition of a sigma-field:
From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
40, COM-417: Lecture 12.2
25.05.2020, 09:38
Sub- and super-martingales, stopping times
35, COM-417: Lecture 11.1
25.05.2020, 09:32
Concentration: large deviations principle
24, COM-417: Lecture 8.1
25.05.2020, 09:13
St Petersburg's paradox and extension of the weak law
Sorry, as the beginning of this one is a bit chaotic; also, it is slightly longer than the others... but it is worth the effort, I think!
APA_lecture11b_partI
11.05.2023, 10:33
9, COM-417: Lecture 3.2
25.05.2020, 08:44
Independence of more sigma-fields
48, COM-417: Lecture 14.1b
24.01.2021, 22:05
Generalization of the MCT to sub- and supermartingales (part b)
APA_lecture12a_partI
17.05.2023, 16:22
9, COM-417: Lecture 3.2
25.05.2020, 08:44
Independence of more sigma-fields
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
APA_lecture14a_partI
31.05.2023, 16:00
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
9, COM-417: Lecture 3.2
25.05.2020, 08:44
Independence of more sigma-fields
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
23, COM-417: Lecture 7.3
25.05.2020, 09:12
Kolmogorov's 0-1 law
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
APA_lecture14b_partIII
01.06.2023, 15:27
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
APA_lecture14b_partII
01.06.2023, 15:27
APA_lecture12a_partII
17.05.2023, 16:23
13, COM-417: Lecture 4.3
25.05.2020, 08:53
Characteristic function
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
42, COM-417: Lecture 13.1
25.05.2020, 09:40
The martingale convergence theorem (version 1)
Note: contrary to what is said at the beginning of the video, this is not the last lecture of the semester: there is still one to come (Lecture 14).
50, COM-417: Lecture 14.2b
25.01.2021, 10:07
McMiarmid's inequality
35, COM-417: Lecture 11.1
25.05.2020, 09:32
Concentration: large deviations principle
APA_lecture13b_partII
25.05.2023, 10:00
2, COM-417: Lecture 1.1
25.05.2020, 08:35
Sigma-fields, sigma-field generated by a collection of events
Here is a small extra comment regarding the definition of a sigma-field:
From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n
APA_lecture12a_partI
17.05.2023, 16:22
10, COM-417: Lecture 3.3
25.05.2020, 08:50
Convolution
39, COM-417: Lecture 12.1
25.05.2020, 09:37
Martingales
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
39, COM-417: Lecture 12.1
25.05.2020, 09:37
Martingales
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
COM-417 Advanced Probability and Applications
15.04.2023, 13:27
APA_lecture14b_partIII
01.06.2023, 15:27
4, COM-417: Lecture 1.3
25.05.2020, 08:38
Sigma-field generated by a collection of random variables
17, COM-417: Lecture 6.1
25.05.2020, 09:00
Cauchy-Schwarz, Jensen and Chebyshev inequalities
27, COM-417: Lecture 9.1
25.05.2020, 09:16
The central limit theorem
APA_lecture13a_partII
24.05.2023, 16:21
22, COM-417: Lecture 7.2 bis
25.05.2020, 09:09
A remark about the proof of the LLN
APA_lecture13a_partI
24.05.2023, 16:21
7, COM-417: Lecture 2.3
25.05.2020, 08:41
The Devil's staircase
26, COM-417: Lecture 8.3
25.05.2020, 09:15
The Curie-Weiss model
32, COM-417: Lecture 10.1b
25.05.2020, 09:28
The coupon collector problem (part 2)
APA recordings (2023)
19.04.2023, 16:28
APA_lecture8a_partI
19.04.2023, 16:20
APA_lecture8a_partII
19.04.2023, 16:27
APA_lecture8b_partI
20.04.2023, 10:38
APA_lecture8b_partII
20.04.2023, 10:39
APA_lecture9b_partI
27.04.2023, 10:22
APA_lecture9b_partII
27.04.2023, 10:23
APA_lecture10b_partI
04.05.2023, 10:37
APA_lecture10b_partII
04.05.2023, 10:37
APA_lecture11b_partI
11.05.2023, 10:33
APA_lecture11b_partII
11.05.2023, 10:34
APA_lecture12a_partI
17.05.2023, 16:22
APA_lecture12a_partII
17.05.2023, 16:23
APA_lecture13a_partI
24.05.2023, 16:21
APA_lecture13a_partII
24.05.2023, 16:21
APA_lecture13b_partI
25.05.2023, 09:59
APA_lecture13b_partII
25.05.2023, 10:00
APA_lecture14a_partI
31.05.2023, 16:00
APA_lecture14a_partII
31.05.2023, 16:00
APA_lecture14b_partI
01.06.2023, 15:27
APA_lecture14b_partII
01.06.2023, 15:27
APA_lecture14b_partIII
01.06.2023, 15:27
APA_lecture14a_partI
31.05.2023, 16:00
17, COM-417: Lecture 6.1
25.05.2020, 09:00
Cauchy-Schwarz, Jensen and Chebyshev inequalities
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
49, COM-417: Lecture 14.2a
25.01.2021, 10:06
Azuma's inequality
3, COM-417: Lecture 1.2
25.05.2020, 08:37
Sub-sigma-fields, random variables
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
6, COM-417: Lecture 2.2
25.05.2020, 08:40
Distribution of a random variable
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
8, COM-417: Lecture 3.1
25.05.2020, 08:43
Independence of two events, random variables, sigma-fields
One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:
- To prove that two random variables are not independent, it suffices indeed to find one counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).
- On the contrary, independence requires that equality holds for all (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
APA_lecture13b_partI
25.05.2023, 09:59
APA_lecture11b_partII
11.05.2023, 10:34
17, COM-417: Lecture 6.1
25.05.2020, 09:00
Cauchy-Schwarz, Jensen and Chebyshev inequalities
35, COM-417: Lecture 11.1
25.05.2020, 09:32
Concentration: large deviations principle
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
4, COM-417: Lecture 1.3
25.05.2020, 08:38
Sigma-field generated by a collection of random variables
APA_lecture10b_partI
04.05.2023, 10:37
18, COM-417: Lecture 6.2
25.05.2020, 09:02
Various types of convergence of sequences of random variables
APA_lecture14b_partIII
01.06.2023, 15:27
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
7, COM-417: Lecture 2.3
25.05.2020, 08:41
The Devil's staircase
APA_lecture14b_partI
01.06.2023, 15:27
APA_lecture14a_partI
31.05.2023, 16:00
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
31, COM-417: Lecture 10.1a
25.05.2020, 09:27
The coupon collector problem (part 1)
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
19, COM-417: Lecture 6.3
25.05.2020, 09:03
Almost sure convergence implies convergence in probability
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
24, COM-417: Lecture 8.1
25.05.2020, 09:13
St Petersburg's paradox and extension of the weak law
Sorry, as the beginning of this one is a bit chaotic; also, it is slightly longer than the others... but it is worth the effort, I think!
APA_lecture13a_partI
24.05.2023, 16:21
13, COM-417: Lecture 4.3
25.05.2020, 08:53
Characteristic function
APA_lecture12a_partII
17.05.2023, 16:23
30, COM-417: Lecture 9.3
25.05.2020, 09:23
Proof of the CLT using characteristic functions
APA recordings (2023)
19.04.2023, 16:28
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
47, COM-417: Lecture 14.1a
24.01.2021, 22:04
Generalization of the MCT to sub- and supermartingales (part a)
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
APA_lecture10b_partII
04.05.2023, 10:37
19, COM-417: Lecture 6.3
25.05.2020, 09:03
Almost sure convergence implies convergence in probability
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
11, COM-417: Lecture 4.1
25.05.2020, 08:52
Expectation of a random variable
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
APA_lecture13a_partII
24.05.2023, 16:21
10, COM-417: Lecture 3.3
25.05.2020, 08:50
Convolution
3, COM-417: Lecture 1.2
25.05.2020, 08:37
Sub-sigma-fields, random variables
APA_lecture10b_partI
04.05.2023, 10:37
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
8, COM-417: Lecture 3.1
25.05.2020, 08:43
Independence of two events, random variables, sigma-fields
One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:
- To prove that two random variables are not independent, it suffices indeed to find one counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).
- On the contrary, independence requires that equality holds for all (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).
24, COM-417: Lecture 8.1
25.05.2020, 09:13
St Petersburg's paradox and extension of the weak law
Sorry, as the beginning of this one is a bit chaotic; also, it is slightly longer than the others... but it is worth the effort, I think!
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
19, COM-417: Lecture 6.3
25.05.2020, 09:03
Almost sure convergence implies convergence in probability
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
17, COM-417: Lecture 6.1
25.05.2020, 09:00
Cauchy-Schwarz, Jensen and Chebyshev inequalities
APA_lecture13b_partI
25.05.2023, 09:59
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
40, COM-417: Lecture 12.2
25.05.2020, 09:38
Sub- and super-martingales, stopping times
41, COM-417: Lecture 12.3
25.05.2020, 09:39
The optional stopping theorem, with a proof
44, COM-417: Lecture 13.2b
25.05.2020, 09:43
Proof of the martingale convergence theorem (part 2)
3, COM-417: Lecture 1.2
25.05.2020, 08:37
Sub-sigma-fields, random variables
13, COM-417: Lecture 4.3
25.05.2020, 08:53
Characteristic function
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
APA_lecture11b_partI
11.05.2023, 10:33
APA_lecture10b_partII
04.05.2023, 10:37
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
41, COM-417: Lecture 12.3
25.05.2020, 09:39
The optional stopping theorem, with a proof
32, COM-417: Lecture 10.1b
25.05.2020, 09:28
The coupon collector problem (part 2)
8, COM-417: Lecture 3.1
25.05.2020, 08:43
Independence of two events, random variables, sigma-fields
One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:
- To prove that two random variables are not independent, it suffices indeed to find one counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).
- On the contrary, independence requires that equality holds for all (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
APA_lecture14b_partII
01.06.2023, 15:27
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
APA_lecture13a_partI
24.05.2023, 16:21
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
31, COM-417: Lecture 10.1a
25.05.2020, 09:27
The coupon collector problem (part 1)
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
31, COM-417: Lecture 10.1a
25.05.2020, 09:27
The coupon collector problem (part 1)
27, COM-417: Lecture 9.1
25.05.2020, 09:16
The central limit theorem
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
11, COM-417: Lecture 4.1
25.05.2020, 08:52
Expectation of a random variable
39, COM-417: Lecture 12.1
25.05.2020, 09:37
Martingales
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
41, COM-417: Lecture 12.3
25.05.2020, 09:39
The optional stopping theorem, with a proof
APA_lecture13b_partI
25.05.2023, 09:59
32, COM-417: Lecture 10.1b
25.05.2020, 09:28
The coupon collector problem (part 2)
APA_lecture13a_partII
24.05.2023, 16:21
22, COM-417: Lecture 7.2 bis
25.05.2020, 09:09
A remark about the proof of the LLN
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
26, COM-417: Lecture 8.3
25.05.2020, 09:15
The Curie-Weiss model
2, COM-417: Lecture 1.1
25.05.2020, 08:35
Sigma-fields, sigma-field generated by a collection of events
Here is a small extra comment regarding the definition of a sigma-field:
From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n
APA_lecture12a_partII
17.05.2023, 16:23
COM-417 Advanced Probability and Applications
15.04.2023, 13:27
24, COM-417: Lecture 8.1
25.05.2020, 09:13
St Petersburg's paradox and extension of the weak law
Sorry, as the beginning of this one is a bit chaotic; also, it is slightly longer than the others... but it is worth the effort, I think!
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
46, COM-417: Lecture 13.3
25.05.2020, 09:44
The martingale convergence theorem (version 2)
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
32, COM-417: Lecture 10.1b
25.05.2020, 09:28
The coupon collector problem (part 2)
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
APA_lecture10b_partII
04.05.2023, 10:37
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
APA recordings (2023)
19.04.2023, 16:28
COM-417 Advanced Probability and Applications
15.04.2023, 13:27
1, COM-417: Introduction and notations
25.05.2020, 08:36
2, COM-417: Lecture 1.1
25.05.2020, 08:35
Sigma-fields, sigma-field generated by a collection of events
Here is a small extra comment regarding the definition of a sigma-field:
From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n
3, COM-417: Lecture 1.2
25.05.2020, 08:37
Sub-sigma-fields, random variables
4, COM-417: Lecture 1.3
25.05.2020, 08:38
Sigma-field generated by a collection of random variables
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
7, COM-417: Lecture 2.3
25.05.2020, 08:41
The Devil's staircase
6, COM-417: Lecture 2.2
25.05.2020, 08:40
Distribution of a random variable
8, COM-417: Lecture 3.1
25.05.2020, 08:43
Independence of two events, random variables, sigma-fields
One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:
- To prove that two random variables are not independent, it suffices indeed to find one counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).
- On the contrary, independence requires that equality holds for all (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).
9, COM-417: Lecture 3.2
25.05.2020, 08:44
Independence of more sigma-fields
10, COM-417: Lecture 3.3
25.05.2020, 08:50
Convolution
11, COM-417: Lecture 4.1
25.05.2020, 08:52
Expectation of a random variable
12, COM-417: Lecture 4.2
25.05.2020, 08:53
Basic properties of expectation
13, COM-417: Lecture 4.3
25.05.2020, 08:53
Characteristic function
14, COM-417: Lecture 5.1
25.05.2020, 08:55
Random vectors
Watch out three small bugs:
- at the beginning of the video: there is of course nothing new here (this was recorded in March 2020)
- around 22:50: (X1,X2) is not a continuous random vector, but it is a random vector
- at the end of the video: the last word written should be "vector" and not "variable"
15, COM-417: Lecture 5.2
25.05.2020, 08:55
Gaussian random vectors
16, COM-417: Lecture 5.3
25.05.2020, 08:57
Joint distribution of a Gaussian random vector
17, COM-417: Lecture 6.1
25.05.2020, 09:00
Cauchy-Schwarz, Jensen and Chebyshev inequalities
18, COM-417: Lecture 6.2
25.05.2020, 09:02
Various types of convergence of sequences of random variables
19, COM-417: Lecture 6.3
25.05.2020, 09:03
Almost sure convergence implies convergence in probability
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
21, COM-417: Lecture 7.2
25.05.2020, 09:08
The law(s) of large numbers, along with its proof
23, COM-417: Lecture 7.3
25.05.2020, 09:12
Kolmogorov's 0-1 law
22, COM-417: Lecture 7.2 bis
25.05.2020, 09:09
A remark about the proof of the LLN
24, COM-417: Lecture 8.1
25.05.2020, 09:13
St Petersburg's paradox and extension of the weak law
Sorry, as the beginning of this one is a bit chaotic; also, it is slightly longer than the others... but it is worth the effort, I think!
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
26, COM-417: Lecture 8.3
25.05.2020, 09:15
The Curie-Weiss model
27, COM-417: Lecture 9.1
25.05.2020, 09:16
The central limit theorem
28, COM-417: Lecture 9.2a
25.05.2020, 09:17
Proof of the CLT via Lindeberg's principle (part 1)
29, COM-417: Lecture 9.2b
25.05.2020, 09:18
Proof of the CLT via Lindeberg's principle (part 2)
30, COM-417: Lecture 9.3
25.05.2020, 09:23
Proof of the CLT using characteristic functions
31, COM-417: Lecture 10.1a
25.05.2020, 09:27
The coupon collector problem (part 1)
33, COM-417: Lecture 10.2
25.05.2020, 09:30
Moments and Carleman's theorem
Here is a link to the Online Encyclopedia of Integer Sequences mentioned at the end of the above video: oeis.org (@Neil Sloane)
32, COM-417: Lecture 10.1b
25.05.2020, 09:28
The coupon collector problem (part 2)
34, COM-417: Lecture 10.3
25.05.2020, 09:31
Concentration: Hoeffding's inequality
35, COM-417: Lecture 11.1
25.05.2020, 09:32
Concentration: large deviations principle
36, COM-417: Lecture 11.2
25.05.2020, 09:33
Conditional expectation: definition
37, COM-417: Lecture 11.3
25.05.2020, 09:34
Basic properties of conditional expectation
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
39, COM-417: Lecture 12.1
25.05.2020, 09:37
Martingales
40, COM-417: Lecture 12.2
25.05.2020, 09:38
Sub- and super-martingales, stopping times
42, COM-417: Lecture 13.1
25.05.2020, 09:40
The martingale convergence theorem (version 1)
Note: contrary to what is said at the beginning of the video, this is not the last lecture of the semester: there is still one to come (Lecture 14).
41, COM-417: Lecture 12.3
25.05.2020, 09:39
The optional stopping theorem, with a proof
43, COM-417: Lecture 13.2a
25.05.2020, 09:42
Proof of the martingale convergence theorem (part 1)
44, COM-417: Lecture 13.2b
25.05.2020, 09:43
Proof of the martingale convergence theorem (part 2)
45, COM-417: Lecture 13.2c
25.05.2020, 09:43
Proof of the martingale convergence theorem (part 3)
46, COM-417: Lecture 13.3
25.05.2020, 09:44
The martingale convergence theorem (version 2)
47, COM-417: Lecture 14.1a
24.01.2021, 22:04
Generalization of the MCT to sub- and supermartingales (part a)
48, COM-417: Lecture 14.1b
24.01.2021, 22:05
Generalization of the MCT to sub- and supermartingales (part b)
49, COM-417: Lecture 14.2a
25.01.2021, 10:06
Azuma's inequality
50, COM-417: Lecture 14.2b
25.01.2021, 10:07
McMiarmid's inequality
51, COM-417 Complement: Some examples of cdfs
03.03.2021, 20:22
52, COM-417 Complement: Monotone class theorem
14.04.2021, 19:03
53, COM-417 Complement: Convergence in distribution
30.04.2021, 11:33
An interesting counter-example
54, COM-417 Complement: Extra lecture on R0
25.05.2020, 10:00
Extra lecture about branching processes, recorded in April 2020
After recording this, I realized it would perhaps be more appropriate to refer to "R" in this lecture, rather than to "R0": R0 is indeed the "basic reproduction number" of the virus, estimated around 3.5, on which one has no control; while R is its "effective reproduction number", on which our behaviour can have a great influence.
APA_lecture11b_partI
11.05.2023, 10:33
APA recordings (2023)
19.04.2023, 16:28
20, COM-417: Lecture 7.1
25.05.2020, 09:04
The Borel-Cantelli lemma
APA_lecture13a_partII
24.05.2023, 16:21
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
APA_lecture13a_partI
24.05.2023, 16:21
27, COM-417: Lecture 9.1
25.05.2020, 09:16
The central limit theorem
10, COM-417: Lecture 3.3
25.05.2020, 08:50
Convolution
19, COM-417: Lecture 6.3
25.05.2020, 09:03
Almost sure convergence implies convergence in probability
APA_lecture14b_partI
01.06.2023, 15:27
34, COM-417: Lecture 10.3
25.05.2020, 09:31
Concentration: Hoeffding's inequality
5, COM-417: Lecture 2.1
25.05.2020, 08:40
Probability measures
APA_lecture13a_partI
24.05.2023, 16:21
APA_lecture11b_partII
11.05.2023, 10:34
38, COM-417: Lecture 11.3 bis
25.05.2020, 09:36
An extra property of conditional expectation
2, COM-417: Lecture 1.1
25.05.2020, 08:35
Sigma-fields, sigma-field generated by a collection of events
Here is a small extra comment regarding the definition of a sigma-field:
From the third axiom regarding countable unions of sets, you can deduce directly that finite unions should also be part of the sigma-field by considering a countable sequence of sets where you replace each A_m by an empty set when m > n; in this case, U_{m>=1} A_m = A_1 U ... U A_n
18, COM-417: Lecture 6.2
25.05.2020, 09:02
Various types of convergence of sequences of random variables
APA_lecture11b_partI
11.05.2023, 10:33
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
APA_lecture11b_partII
11.05.2023, 10:34
25, COM-417: Lecture 8.2
25.05.2020, 09:14
Convergence in distribution
16, COM-417: Lecture 5.3
25.05.2020, 08:57
Joint distribution of a Gaussian random vector
Lectures:
- In-person in the room DIA 004 on Wed 1-4 PM and ELG 120 on Thu 9-12 AM.
- Thursday lectures will take place only on select weeks. Other weeks Thursdays are reserved for the exercise session. This will be clearly marked each week on Moodle.
Exercise Sessions:
- In-person in the room ELG 120 on Thu 9-12 PM on select weeks. This will be clearly marked each week on Moodle.
- Problem sets will be posted for each unit covered in lecture. You are expected to work on them outside of class and during exercise sessions. Problem sets will not be graded. However, it is important that you do them regularly if you would like to succeed in the course.
Grading Scheme:
- Midterm exam #1 - 25%
- Midterm exam #2 - 25%
- Final exam - 50%
Midterm Exam #1: Wednesday, October 8, 1:15pm - 4pm, room DIA 004 and AAC 0 08.
- Allowed material: one cheat sheet (i.e., two single-sided A4 handwritten pages).
Midterm Exam #2: Wednesday, November 26, 1:15pm - 4pm, AAC 2 31.
- Allowed material: two cheat sheets (i.e., four single-sided A4 handwritten pages). You may re-use the cheatsheet from midterm 1, and create a new additional page. Or, create two pages from scratch.
Final Exam: Wednesday, January 21, 2026, 9:15am - 12:15pm, room CM 1 121.
- Allowed material: two cheat sheets (i.e., four single-sided A4 handwritten pages).
- Please note that the exam content will focus more on the part of the course not covered on the midterms, but will also cover material already covered by the midterms.
Prof. Yanina Shkel || INR 131 || yanina.shkel@epfl.ch
- Terence Tao, An Introduction to Measure Theory, Preprint, Softcover ISBN: 978-1-4704-6640-4
- Sheldon M. Ross, Erol A. Pekoz, A Second Course in Probability, 1st edition, 2007.
- Jeffrey S. Rosenthal, A First Look at Rigorous Probability Theory, 2nd edition, World Scientific, 2006.
- Geoffrey R. Grimmett, David R. Stirzaker, Probability and Random Processes, 3rd edition, Oxford University Press, 2001.
- Sheldon M. Ross, Stochastic Processes, 2nd edition, Wiley, 1996.
- William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1&2, Wiley, 1950.
- (more advanced) Richard Durrett, Probability: Theory and Examples, 4th edition, Cambridge University Press, 2010.
- (more advanced) Patrick Billingsley, Probability and Measure, 3rd edition, Wiley, 1995.
- Announcements (Forum)
- Lecture notes (File)
- Extra material: Fubini and Radon-Nikodym (File)
- Sample Midterm (Fall 2024) (File)
- Sample Midterm Solutions (Fall 2024) (File)
- Sample Midterm (Spring 2024) (File)
- Sample Midterm Solutions (Spring 2024) (File)
- Sample Midterm (Spring 2023) (File)
- Sample Midterm Solutions (Spring 2023) (File)
- Sample Final (Fall 2024) (File)
- Sample Final Solutions (Fall 2024) (File)
- Sample Final (Spring 2024) (File)
- Sample Final Solutions (Spring 2024) (File)
- Extra Problems (Folder)
- Final Exam Solutions (File)
Week 1 (September 10-11)
Wed (lecture): Sigma-fields and random variables (chapter 1); probability measures (section 2.1)
Thu (lecture): Probability measures and distributions (section 2.1-2.5)
Corresponding Videos
Week 2 (September 17-18)
Wed (lecture): Cantor set and the devil's staircase (section 2.5); independence (section 3.1-3.5)
Thu (exercises): Problem sets 1 and 2;
Corresponding Videos
Week 3 (September 24-25)
Wed (lecture): expectation (chapter 4)
Thu (exercises): Problem sets 3 and 4
Corresponding Videos
Week 4 (October 1-2)
Wed (lecture): Probability couplings (chapter 5)
Thu (exercises): Review problem sets 1-4, problem set 5
No Corresponding Videos - see lecture notes
Week 5 (October 8-9) MIDTERM #1
Thu (exercises): Review of midterm exam #1
Midterm Exam #1 Statistics: (over 50)Mean: 31.34
Median: 32
Min / Max: 8 / 48.5
Week 6 (October 15-16)
Thu (lecture): Inequalities (chapter 6); transform methods (chapter 7);
Corresponding Videos
- Problem Set 5 (File)
- Solution Set 5 (File)
- Problem Set 6 (File)
- Solution Set 6 (File)
- Problem Set 7 (File)
- Solution Set 7 (File)
Just for fun: holiday quiz
- Quiz 1 (Choice)
- Quiz 2 (Choice)
- Quiz 3 (Choice)
- Holiday quiz solutions (File)
- And here is yet another quiz... (URL)
Week 7 (October 29-30)
Thu (exercises): Work on problem sets 5, 6, and 7
Corresponding Videos
- moments and moment generating function (no video available: see lecture notes and problem set 7, exercise 2)
- random vectors; Gaussian random vectors; joint distribution of Gaussian random vectors
Week 8 (November 5-6)
Thu (lecture): Laws of large numbers - weak and strong, proof (sections 9.4-9.6); Kolmogorov's 0-1 law
Corresponding Videos
- convergences of random variables; almost sure convergence vs convergence in probability
- Borrel-Contalli lemma; laws of large numbers; addendum
- Kolmogorov’s 0-1 law
Week 9 (November 12-13)
Thu (exercises): Work on problem sets 8 and 9
Corresponding Videos
- St-Petersburg paradox
- Convergence in distribution; equivalent definition of convergence in distribution
- The Central Limit Theorem; proof of CLT; alternative proof of the CLT;
- Application: coupon collector problem 1; and 2
Note : In chapter 10 we will NOT cover the Curie-Weiss model (section 10.2) and the Lindeberg’s principle (part of section 10.4).
Week 10 (November 19-20)
Thu (exercises): Work on problem sets 10 and 11
Corresponding Videos
- Conditional expectation, properties, more properties
- See also from 2023: conditional expectation and properties,
Week 11 (November 26-27) MIDTERM #2
Thu (lecture): Conditional expectation (chapter 11); Martingales (chapter 12)
Midterm #2 covers the material up to chapter 10 and problem set 10. In particular, conditional expectation and problem set 11 will NOT be on the second midterm
Midterm Exam #2 Statistics: (over 50)
Mean: 31
Median: 31
Min / Max: 4.5 / 45
Corresponding Videos
Week 12 (December 3-4)
Thu (exercises): Midterm #2 review
We will NOT cover section 12.4, The reflection principle.
Corresponding Videos
- MCT v1;
- From 2023 - Martingale transform; Doob's decomposition theorem (Brownian motion part is optional); Doob's martingale;
Week 13 (December 10-11)
Thu (exercises): Work on problem sets 11 and 12
Corresponding Videos
- MCT proof part 1; MCT proof part 2; MCT proof part 3 (focus on maximal inequality)
- MCT v2; Generalizations of MCT part a); Generalizations of MCT part b);
- From 2023 - Consequences of MCT; MCT proof part 1; MCT proof part 2 (focus on maximal inequality); MCT v2; Generalizations
Week 14 (December 17-18)
Thu (exercises): Work on problem sets 13 and 14
Corresponding Videos