STAT/MATH 395 PROBABILITY II

Transcription

1 STAT/MATH 395 PROBABILITY II Bivariate Distributions Néhémy Lim University of Washington Winter 2017

2 Outline Distributions of Two Random Variables Distributions of Two Discrete Random Variables Distributions of Two Continuous Random Variables The Correlation Coefficient Covariance Correlation Conditional Distributions Discrete case Continuous case

3 Outline Distributions of Two Random Variables Distributions of Two Discrete Random Variables Distributions of Two Continuous Random Variables The Correlation Coefficient Covariance Correlation Conditional Distributions Discrete case Continuous case

4 Example: Climate Change

5 Objectives Extend the definition of a probability distribution of one random variable to the joint probability distribution of two random variables Learn a way of quantifying the extent to which two random variables are related Define the conditional probability distribution of a random variable X given that Y has occurred

6 Definition Let X and Y be two rrvs on probability space (Ω, A, P). A two-dimensional random variable (X, Y ) is a function mapping (X, Y ) : Ω R 2, such that for any numbers x, y R: {ω Ω X(ω) x and Y (ω) y} A (1) Definition (Joint cumulative distribution function) Let X and Y be two rrvs on probability space (Ω, A, P). The joint (cumulative) distribution function (joint c.d.f.) of X and Y is the function F XY given by F XY (x, y) = P ({X x} {Y y}) P (X x, Y y), (2) for x, y R

7 Why Gen Y Loves Restaurants And Restaurants Love Them Even More According to a new report from the research firm Technomic, 42% of millennials say they visit upscale casual-dining restaurants at least once a month. That s a higher percentage than Gen X (33%) and Baby Boomers (24%) who go to such restaurants once or more monthly. Time, Aug. 15, 2012, By Brad Tuttle Age Population ,267, (Millenials) 62,649, (Gen X) 63,779, (Baby Boomers) 71,216, ,832,721 Demography in the US by age group [US Census data]

8 Definition Let X and Y be two discrete rrvs on probability space (Ω, A, P). The joint pmf of X and Y, denoted by p XY, is defined as follows : p XY (x, y) = P ({X = x} {Y = y}) P (X = x, Y = y), (3) for x X(Ω) and y Y (Ω)

9 Why Gen Y Loves Restaurants And Restaurants Love Them Even More X = 1 : the person visits upscale restaurants at least once a month / {X = 0} = {X = 1} c Y = 1 : the person is a millenial / Y = 2 : the person is a Gen X / Y = 3 : the person is a Baby Boomer X Y X Y ,337 42,732 47, ,313 21,047 24,213 Table 1: Count Data ( 1000) Table 2: Joint Probability Mass Function p XY (x, y)

10 Property Let X and Y be two discrete rrvs on probability space (Ω, A, P) with joint pmf p XY, then the following holds : p XY (x, y) 0, for x X(Ω) and y Y (Ω) x X(Ω) y Y (Ω) p XY (x, y) = 1 Property Let X and Y be two disctete rrvs on probability space (Ω, A, P) with joint pmf p XY. Then, for any subset B X(Ω) Y (Ω) P ((X, Y ) B) = (x,y) B p XY (x, y)

11 Definition (Marginal distributions) Let X and Y be two discrete rrvs on probability space (Ω, A, P) with joint pmf p XY. Then the pmf of X alone is called the marginal probability mass function of X and is defined by: p X (x) = P(X = x) = y Y (Ω) p XY (x, y), for x X(Ω) (4) Similarly, the pmf of Y alone is called the marginal probability mass function of Y and is defined by: p Y (y) = P(Y = y) = x X(Ω) p XY (x, y), for y Y (Ω) (5)

12 Definition (Independence) Let X and Y be two discrete rrvs on probability space (Ω, A, P) with joint pmf p XY. Let p X and p Y be the respective marginal pmfs of X and Y. Then X and Y are said to be independent if and only if : p XY (x, y) = p X (x)p Y (y), for all x X(Ω) and y Y (Ω) (6)

13 Definition (Expected Value) Let X and Y be two discrete rrvs on probability space (Ω, A, P) with joint pmf p XY and let g : R 2 R be a bounded piecewise continuous function. Then, the mathematical expectation of g(x, Y ), if it exists, is denoted by E[g(X, Y )] and is defined as follows : E[g(X, Y )] = x X(Ω) y Y (Ω) g(x, y)p XY (x, y) (7)

14 Example Consider the following joint probability mass function : p XY (x, y) = xy S(x, y) with S = {(1, 1), (1, 2), (2, 2)} 1. Show that p XY is a valid joint probability mass function. 2. What is P(X + Y 3)? 3. Give the marginal probability mass functions of X and Y. 4. What are the expected values of X and Y? 5. Are X and Y independent?

15 Definition (Joint probability density function) Let X and Y be two continuous rrvs on probability space (Ω, A, P). X and Y are said to be jointly continuous if there exists a function f XY such that, for any Borel set on R 2 : P((X, Y ) B) = f XY (x, y) dx dy (8) Then function f XY is called the joint probability density function of X and Y. Moreover, if the joint distribution function F XY is of class C 2, then the joint pdf of X and Y can be expressed in terms of partial derivatives : f XY (x, y) = 2 F (x, y) (9) x y B

16 Property Let X and Y be two continuous rrvs on probability space (Ω, A, P) with joint pdf f XY, then the following holds : f XY (x, y) 0, for x, y R f XY (x, y) dx dy = 1 Example. Let X and Y be two continuous random variables with joint probability density function : f XY (x, y) = 4xy1 [0,1] 2(x, y) Verify that f XY is a valid joint probability density function. What is P(Y < X)?

17 Definition (Marginal distributions) Let X and Y be two continuous rrvs on probability space (Ω, A, P) with joint pdf f XY. Then the pdf of X alone is called the marginal probability density function of X and is defined by: f X (x) = f XY (x, y) dy, for x R (10) Similarly, the pdf of Y alone is called the marginal probability density function of Y and is defined by: f Y (y) = f XY (x, y) dx, for y R (11) Example. Let X and Y be two continuous random variables with joint probability density function : f XY (x, y) = 4xy1 [0,1] 2(x, y) What are the marginal probability density functions of X and Y?

18 Definition (Independence) Let X and Y be two continuous rrvs on probability space (Ω, A, P) with joint pdf f XY. Let f X and f Y be the respective marginal pdfs of X and Y. Then X and Y are said to be independent if and only if : f XY (x, y) = f X (x)f Y (y), for all x, y R (12) Example. Let X and Y be two continuous random variables with joint probability density function : Are X and Y independent? f XY (x, y) = 4xy1 [0,1] 2(x, y)

19 Definition (Expected Value) Let X and Y be two continuous rrvs on probability space (Ω, A, P) with joint pdf f XY and let g : R 2 R be a bounded piecewise continuous function. Then, the mathematical expectation of g(x, Y ), if it exists, is denoted by E[g(X, Y )] and is defined as follows : E[g(X, Y )] = g(x, y)f XY (x, y) dx dy (13) Example. Let X and Y be two continuous random variables with joint probability density function : f XY (x, y) = 4xy1 [0,1] 2(x, y) What are the expected values of X and Y?

20 Example The joint probability density function of X and Y is given by: f XY (x, y) = 6 ( x 2 + xy ) 1 S (x, y) 7 2 with S = {(x, y) 0 < x < 1, 0 < y < 2} 1. Show that f XY is a valid joint probability density function. 2. Compute the marginal probability density function of X. 3. Find P(X > Y ). 4. What are the expected values of X and Y? 5. Are X and Y independent?

21 Outline Distributions of Two Random Variables Distributions of Two Discrete Random Variables Distributions of Two Continuous Random Variables The Correlation Coefficient Covariance Correlation Conditional Distributions Discrete case Continuous case

22 Definition (Covariance) Let X and Y be two rrvs on probability space (Ω, A, P). The covariance of X and Y, denoted by Cov(X, Y ), is defined as follows : Cov(X, Y ) = E[(X E[X])(Y E[Y ])] (14) upon existence of the above expression If X and Y are discrete rrv with joint pmf p XY, then the covariance of X and Y is : Cov(X, Y ) = x X(Ω) y Y (Ω) (x E[X])(y E[Y ])p XY (x, y) If X and Y are continuous rrv with joint pdf f XY, then the covariance of X and Y is : Cov(X, Y ) = (15) (x E[X])(y E[Y ])f XY (x, y) dx dy (16)

23 Theorem Let X and Y be two rrvs on probability space (Ω, A, P). The covariance of X and Y can be calculated as : Cov(X, Y ) = E[XY ] E[X] E[Y ] (17) Example 6. Suppose that X and Y have the following joint probability mass function: X Y What is the covariance of X and Y?

24 Property Here are some properties of the covariance. For any random variables X and Y, we have : 1. Cov(X, Y ) = Cov(Y, X) 2. Cov(X, X) = Var(X) 3. Cov(aX, Y ) = acov(x, Y ) for a R 4. Let X 1,..., X n be n random variables and Y 1,..., Y m be m random variables. Then : n m n m Cov X i, = Cov(X i, Y j ) i=1 Y j j=1 i=1 j=1

25 Definition (Correlation) Let X and Y be two rrvs on probability space (Ω, A, P) with respective standard deviations σ X = Var(X) and σ Y = Var(Y ). The correlation of X and Y, denoted by ρ XY, is defined as follows : ρ XY = Cov(X, Y ) σ X σ Y (18) upon existence of the above expression Property Let X and Y be two rrvs on probability space (Ω, A, P). 1 ρ XY 1 (19)

26 Interpretation of Correlation The correlation coefficient of X and Y is interpreted as follows: If ρ XY = 1, then X and Y are perfectly, positively, linearly correlated. If ρ XY = 1, then X and Y are perfectly, negatively, linearly correlated. X and Y are also said to be perfectly linearly anticorrelated. If ρ XY = 0, then X and Y are completely, linearly uncorrelated. If 0 < ρ XY < 1, then X and Y are positively, linearly correlated. If 1 < ρ XY < 0, then X and Y are negatively, linearly correlated.

27 Example : Dining Habits X = 1 : a person between 20 and 69 visits upscale restaurants at least once a month and X = 0 otherwise Y = 1 : a person between 20 and 69 is a millenial Y = 2 : a person between 20 and 69 is a Gen X Y = 3 : a person between 20 and 69 is a Baby Boomer The joint probability mass function of X and Y is given below : X Y Compute the covariance and the correlation of X and Y. What can you say about the relationship between X and Y?

28 Correlation is not causation!

29 Theorem Let X and Y be two rrvs on probability space (Ω, A, P) and let g 1 and g 2 be two bounded piecewise continuous functions. If X and Y are independent, then E[g 1 (X)g 2 (Y )] = E[g 1 (X)] E[g 2 (Y )] (20) provided that the expectations exist. Theorem Let X and Y be two rrvs on probability space (Ω, A, P). If X and Y are independent, then Cov(X, Y ) = ρ XY = 0 (21)

30 Example Let X and Y be discrete random variables with joint pmf: X Y What is the correlation between X and Y? 2. Are X and Y independent?

31 Outline Distributions of Two Random Variables Distributions of Two Discrete Random Variables Distributions of Two Continuous Random Variables The Correlation Coefficient Covariance Correlation Conditional Distributions Discrete case Continuous case

32 Example An international TV network company is interested in the relationship between the region of citizenship of its customers and their favorite sport. Sports Citizenship Africa America Asia Europe Tennis Basketball Soccer Football

33 Definition Let X and Y be two discrete rrvs on probability space (Ω, A, P) with joint pmf p XY and respective marginal pmfs p X and p Y. Then, for y Y (Ω), the conditional probability mass function of X given Y = y is defined by : p X Y (x y) = P(X = x Y = y) = p XY (x, y) p Y (y) (22) provided that p Y (y) 0.

34 Definition (Conditional Expectation) Let X and Y be two discrete rrvs on probability space (Ω, A, P). Then, for y Y (Ω), the conditional expectation of X given Y = y is defined as follows : E[X Y = y] = x X(Ω) xp X Y (x y) (23) Property (Linearity of conditional expectation) Let X and Y be two rrvs on probability space (Ω, A, P). If c 1, c 2 R and g 1 : X(Ω) R and g 2 : X(Ω) R are piecewise continuous functions. Then, for y Y (Ω), we have : E[c 1 g 1 (X)+c 2 g 2 (X) Y = y] = c 1 E[g 1 (X) Y = y]+c 2 E[g 2 (X) Y = y] (24)

35 Property Let X and Y be two discrete rrvs on probability space (Ω, A, P). If X and Y are independent, then, for y Y (Ω), we have : p X Y (x y) = p X (x) for all x X(Ω) (25) Similarly, for x X(Ω), we have : Also, p Y X (y x) = p Y (y) for all y Y (Ω) (26) Similarly, for x X(Ω), we have : E[X Y = y] = E[X] (27) E[Y X = x] = E[Y ] (28) Remark : Those properties are also true for continuous rrvs.

36 Definition (Conditional Variance) Let X and Y be two rrvs on probability space (Ω, A, P). Then, for y Y (Ω), the conditional variance of X given Y = y is defined as follows : Var(X Y = y) = E[(X E[X Y = y]) 2 Y = y] (29) Property For y Y (Ω), the conditional variance of X given Y = y can be calculated as follows : Var(X Y = y) = E[X 2 Y = y] (E[X Y = y]) 2 (30)

37 Definition Let X and Y be two continuous rrvs on probability space (Ω, A, P) with joint pdf f XY and respective marginal pdfs f X and f Y. Then, for y R, the conditional probability density function of X given Y = y is defined by : f X Y (x y) = f XY (x, y) f Y (y) (31) provided that f Y (y) 0.

38 Definition (Conditional Expectation) Let X and Y be two continuous rrvs on probability space (Ω, A, P). Then, for y R, the conditional expectation of X given Y = y is defined as follows : E[X Y = y] = xf X Y (x y) dx (32)

39 Example Let X and Y be two continuous random variables with joint probability density function : f XY (x, y) = S(x, y) with S = {(x, y) x 2 < y < 1, 0 < x < 1} Compute the expected value of Y given X = x.