www.stat.ubc.ca/~bouchard/courses/stat302-sp2017-18/ Intro to Probability Instructor: Aleandre Bouchard
Announcements New webwork will be release by end of day today, due one week later.
Plan for today Intro to multivariate distributions Joint distributions Marginal distributions Independence of continuous random variables
Joint densities
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?
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
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
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.
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)
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?
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
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 )
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
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
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
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
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
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
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
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
Marginal distributions
Def 24 Eample of marginal densities Marginal of X 0.6 Marginal fx() density 0.4 0.2 0.0 1.0 0.5 0.0 0.5 1.0 of Y fy(y) Height of the marginal at 1.0 1.0 = 0 obtained by 0.5 0.5 integrating the joint density over y at = 0: Y 0.0 y 0.0 f X () = Z +1 1 f(, y) dy 0.5 1.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 X 0.0 0.2 0.4 0.6 density
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. 0.45 B. 0.53 f X () = Z +1 1 f(, y) dy C. 0.55 D. 0.63 y f(, y) = 1 B(, y) area(b) B = {(, y) : 2 + y 2 = 1} area(b) = π
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. 0.45 B. 0.53 f X () = Z +1 1 f(, y) dy C. 0.55 D. 0.63 y f(, y) = 1 B(, y) area(b) B = {(, y) : 2 + y 2 = 1} area(b) = π
Independence vs. dependence for continuous random variables
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)
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)
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 )
Eamples of nonuniform joint density
E 67a Eample Suppose (X,Y) has joint density: f(, y) =2e 2y for > 0 and y > 0 P(X > 1, Y < 1)?
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