Advanced probability and applications

16, COM-417: Lecture 5.3

25.05.2020, 08:57

Joint distribution of a Gaussian random vector

APA_lecture12a_partI

17.05.2023, 16:22

7, COM-417: Lecture 2.3

25.05.2020, 08:41

The Devil's staircase

5, COM-417: Lecture 2.1

25.05.2020, 08:40

Probability measures

6, COM-417: Lecture 2.2

25.05.2020, 08:40

Distribution of a random variable

APA_lecture13b_partII

25.05.2023, 10:00

13, COM-417: Lecture 4.3

25.05.2020, 08:53

Characteristic function

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)

29, COM-417: Lecture 9.2b

25.05.2020, 09:18

Proof of the CLT via Lindeberg's principle (part 2)

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

9, COM-417: Lecture 3.2

25.05.2020, 08:44

Independence of more sigma-fields

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

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

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 

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

21, COM-417: Lecture 7.2

25.05.2020, 09:08

The law(s) of large numbers, along with its proof

25, COM-417: Lecture 8.2

25.05.2020, 09:14

Convergence in distribution

APA_lecture14b_partII

01.06.2023, 15:27

APA_lecture12a_partII

17.05.2023, 16:23

12, COM-417: Lecture 4.2

25.05.2020, 08:53

Basic properties of expectation

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"

15, COM-417: Lecture 5.2

25.05.2020, 08:55

Gaussian random vectors

APA_lecture14b_partIII

01.06.2023, 15:27

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

17, COM-417: Lecture 6.1

25.05.2020, 09:00

Cauchy-Schwarz, Jensen and Chebyshev inequalities

15, COM-417: Lecture 5.2

25.05.2020, 08:55

Gaussian random vectors

18, COM-417: Lecture 6.2

25.05.2020, 09:02

Various types of convergence of sequences of random variables

APA_lecture14b_partI

01.06.2023, 15:27

APA_lecture14a_partI

31.05.2023, 16:00

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!

11, COM-417: Lecture 4.1

25.05.2020, 08:52

Expectation of a random variable

APA_lecture13a_partII

24.05.2023, 16:21

10, COM-417: Lecture 3.3

25.05.2020, 08:50

Convolution

APA_lecture10b_partI

04.05.2023, 10:37

12, COM-417: Lecture 4.2

25.05.2020, 08:53

Basic properties of expectation

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

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

APA_lecture13b_partI

25.05.2023, 09:59

41, COM-417: Lecture 12.3

25.05.2020, 09:39

The optional stopping theorem, with a proof

36, COM-417: Lecture 11.2

25.05.2020, 09:33

Conditional expectation: definition

25, COM-417: Lecture 8.2

25.05.2020, 09:14

Convergence in distribution

APA_lecture13a_partI

24.05.2023, 16:21

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)

20, COM-417: Lecture 7.1

25.05.2020, 09:04

The Borel-Cantelli lemma

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)

22, COM-417: Lecture 7.2 bis

25.05.2020, 09:09

A remark about the proof of the LLN

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)

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)

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

38, COM-417: Lecture 11.3 bis

25.05.2020, 09:36

An extra property of conditional expectation

27, COM-417: Lecture 9.1

25.05.2020, 09:16

The central limit theorem

APA_lecture13a_partI

24.05.2023, 16:21

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_lecture11b_partI

11.05.2023, 10:33

APA_lecture11b_partII

11.05.2023, 10:34

16, COM-417: Lecture 5.3

25.05.2020, 08:57

Joint distribution of a Gaussian random vector

APA_lecture12a_partI

17.05.2023, 16:22

7, COM-417: Lecture 2.3

25.05.2020, 08:41

The Devil's staircase

5, COM-417: Lecture 2.1

25.05.2020, 08:40

Probability measures

6, COM-417: Lecture 2.2

25.05.2020, 08:40

Distribution of a random variable

APA_lecture13b_partII

25.05.2023, 10:00

13, COM-417: Lecture 4.3

25.05.2020, 08:53

Characteristic function

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)

29, COM-417: Lecture 9.2b

25.05.2020, 09:18

Proof of the CLT via Lindeberg's principle (part 2)

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

9, COM-417: Lecture 3.2

25.05.2020, 08:44

Independence of more sigma-fields

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

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

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 

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

21, COM-417: Lecture 7.2

25.05.2020, 09:08

The law(s) of large numbers, along with its proof

25, COM-417: Lecture 8.2

25.05.2020, 09:14

Convergence in distribution

APA_lecture14b_partII

01.06.2023, 15:27

APA_lecture12a_partII

17.05.2023, 16:23

12, COM-417: Lecture 4.2

25.05.2020, 08:53

Basic properties of expectation

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"

15, COM-417: Lecture 5.2

25.05.2020, 08:55

Gaussian random vectors

APA_lecture14b_partIII

01.06.2023, 15:27

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

17, COM-417: Lecture 6.1

25.05.2020, 09:00

Cauchy-Schwarz, Jensen and Chebyshev inequalities

15, COM-417: Lecture 5.2

25.05.2020, 08:55

Gaussian random vectors

18, COM-417: Lecture 6.2

25.05.2020, 09:02

Various types of convergence of sequences of random variables

APA_lecture14b_partI

01.06.2023, 15:27

APA_lecture14a_partI

31.05.2023, 16:00

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!

11, COM-417: Lecture 4.1

25.05.2020, 08:52

Expectation of a random variable

APA_lecture13a_partII

24.05.2023, 16:21

10, COM-417: Lecture 3.3

25.05.2020, 08:50

Convolution

APA_lecture10b_partI

04.05.2023, 10:37

12, COM-417: Lecture 4.2

25.05.2020, 08:53

Basic properties of expectation

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

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

APA_lecture13b_partI

25.05.2023, 09:59

41, COM-417: Lecture 12.3

25.05.2020, 09:39

The optional stopping theorem, with a proof

36, COM-417: Lecture 11.2

25.05.2020, 09:33

Conditional expectation: definition

25, COM-417: Lecture 8.2

25.05.2020, 09:14

Convergence in distribution

APA_lecture13a_partI

24.05.2023, 16:21

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)

20, COM-417: Lecture 7.1

25.05.2020, 09:04

The Borel-Cantelli lemma

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)

22, COM-417: Lecture 7.2 bis

25.05.2020, 09:09

A remark about the proof of the LLN

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)

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)

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

38, COM-417: Lecture 11.3 bis

25.05.2020, 09:36

An extra property of conditional expectation

27, COM-417: Lecture 9.1

25.05.2020, 09:16

The central limit theorem

APA_lecture13a_partI

24.05.2023, 16:21

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_lecture11b_partI

11.05.2023, 10:33

APA_lecture11b_partII

11.05.2023, 10:34