Intro to Probability Instructor: Alexandre Bouchard

Transcription

1 Intro to Probability Instructor: Aleandre Bouchard

2 Announcements New webwork will be release by end of day today, due one week later.

3 Plan for today Intro to multivariate distributions Joint distributions Marginal distributions Independence of continuous random variables

4 Joint densities

5 E 59 Motivating problem A man and a woman try to meet at a certain place between 1:00pm and 2:00pm. Suppose each person pick an arrival time between 1:00pm and 2:00pm uniformly at random, and waits for the other at most 10 minutes. What is the probability that they meet?

6 Recall: density (for one random variable) The function f() is a density for X if for any subset A of the real line: Eample: A = [a, b] area = probability height = density

7 Def 23 Today: density (for two random variables) The function f(, y) is a joint density for X, Y if for any subset A of the plane: Z P ((X, Y ) 2 A) = (,y)2a f(, y) d dy volume = probability Eample: A = [a, b] [c, d] height = density Notation for rectangle with one side equal to [a, b] and the other equal to [c,d] b a c d y

8 E 60 Eample: uniform density on a subset B of the plane Eample: B y density height = density = 1/ area(b) y f(, y) = 1 B(, y) area(b) Recall: 1 B (, y) = 1 if (, y) 2 B 0 o.w.

9 E 61 Eample: uniform density on a subset B of the plane Another eample: B y density height = density = 1/ area(b) y f(, y) = 1 B(, y) area(b)

10 E 62 Simpler eample first Random variables: X: time a woman arrives at meeting (min) Y: time a man arrives at meeting (min) Distribution: (X, Y) uniform on the square, [0min, 60min] [0min, 60min] Probability they both arrive in the first 30 minutes?

11 E 62 Computing probabilities from joint densities: eample Random variables: X: time a woman arrives at meeting (min) Y: time a man arrives at meeting (min) Distribution: (X, Y) uniform on the square, [0min, 60min] [0min, 60min] Probability they both arrive in the first 30 minutes? f(, y) = 1 B(, y) area(b) support B y 60 P ((X, Y ) 2 A) = Z (,y)2a f(, y) d dy A 30 query 30 60

12 E 63 Compute the following probabilities Let X, Y be uniform on [0, 1] [0, 1] For each of these, draw the query area and compute the corresponding probability: P( X - Y < 1/2 ) P( X/Y - 1 < 1/2 ) P( Y > X Y > 1/2 )

13 E 63a Clicker questions Let X, Y be uniform on [0, 1] [0, 1] Query area of P( X - Y < 1/2 )? y A B y y C D None of these

14 E 63a Clicker questions Let X, Y be uniform on [0, 1] [0, 1] Query area of P( X - Y < 1/2 )? y A B y y C D None of these

15 E 63b Clicker questions Let X, Y be uniform on [0, 1] [0, 1] P( X - Y < 1/2 )? A. 1/4 B. 1/2 C. 3/4 D. 4/5

16 E 63b Clicker questions Let X, Y be uniform on [0, 1] [0, 1] P( X - Y < 1/2 )? A. 1/4 B. 1/2 C. 3/4 D. 4/5

17 E 63c Clicker questions Let X, Y be uniform on [0, 1] [0, 1] P( X/Y - 1 < 1/2 ) A. 1/5 B. 5/12 C. 7/12 D. 4/5

18 E 63c Clicker questions Let X, Y be uniform on [0, 1] [0, 1] P( X/Y - 1 < 1/2 ) A. 1/5 B. 5/12 C. 7/12 D. 4/5

19 E 63d Clicker questions Let X, Y be uniform on [0, 1] [0, 1] P( Y > X Y > 1/2 ) A. 3/4 B. 4/5 C. 4/7 D. 4/9

20 E 63d Clicker questions Let X, Y be uniform on [0, 1] [0, 1] P( Y > X Y > 1/2 ) A. 3/4 B. 4/5 C. 4/7 D. 4/9

21 Marginal distributions

22 Def 24 Eample of marginal densities Marginal of X 0.6 Marginal fx() density of Y fy(y) Height of the marginal at = 0 obtained by integrating the joint density over y at = 0: Y 0.0 y 0.0 f X () = Z +1 1 f(, y) dy X density

23 E 64 Eercise: computing a marginal density If (X, Y) is uniform on the circle, find fx(0.5) Height of the marginal at obtained by integrating the joint density over y at : A B f X () = Z +1 1 f(, y) dy C D y f(, y) = 1 B(, y) area(b) B = {(, y) : 2 + y 2 = 1} area(b) = π

24 E 64 Eercise: computing a marginal density If (X, Y) is uniform on the circle, find fx(0.5) Height of the marginal at obtained by integrating the joint density over y at : A B f X () = Z +1 1 f(, y) dy C D y f(, y) = 1 B(, y) area(b) B = {(, y) : 2 + y 2 = 1} area(b) = π

25 Independence vs. dependence for continuous random variables

26 Def 25 Equivalent definitions X and Y are independent Useful to show that r.v. s are NOT indep For all intervals, A1, A2: P (X 2 A 1,Y 2 A 2 )=P(X 2 A 1 )P (Y 2 A 2 ) Useful to show that r.v. s are indep The joint density of (X, Y) can be written as: f(, y) =h()k(y)

27 E 65 Eample: two random variables that are independent y why? d The joint density of (X, Y) can be written as: c f(, y) =h()k(y) a b h() k(y) f(, y) = 1 B(, y) area(b) = 1[a,b] () area(b) 1 [c,d] (y)

28 E 66 Eample: two random variables that are NOT independent why? y A2 For some intervals, A1, A2: P (X 2 A 1,Y 2 A 2 )=P (X 2 A 1 )P (Y 2 A 2 ) A1 Pick A1, A2 as shown on the left Which one(s) of these are zero? (use material from earlier today) P (X 2 A 1,Y 2 A 2 ) P (X 2 A 1 )P )P (Y 2 A 2 )

29 Eamples of nonuniform joint density

30 E 67a Eample Suppose (X,Y) has joint density: f(, y) =2e 2y for > 0 and y > 0 P(X > 1, Y < 1)?

31 E 67b Eample Suppose (X,Y) has joint density: f(, y) =2e 2y for > 0 and y > 0 P(X > 1, Y < 1)? P(X < Y)? X indep of Y? e -1 (1 - e -2 ) A. 1/2 B. 1/3 C. 1/4 D. 1/5