Chap 8 Bivariate Distributions Ghahramani 3rd edition 2019/2/24

Outline 8.1 Joint distribution of two random variables 8.2 Independent random variables 8.3 Conditional distributions 8.4 Transformations of two random variables

8.1 Joint distributions of 2 r.v.’s (Joint probability mass functions) Def Let X and Y be 2 discrete random variables defined on the same sample space. Let the sets of possible values of X and Y be A and B, respectively. The function p(x, y)=P(X=x, Y=y) is called the joint probability mass function of X and Y.

Joint distributions of 2 r.v.’s Def Let X and Y have joint probability function p(x,y). Let the sets of possible values of X and Y be A and B, respectively. Then the functions are called, respectively, the marginal probability mass functions of X and Y.

Joint distributions of 2 r.v.’s Ex 8.1 A small college has 90 male and 30 female professors. An ad hoc committee of 5 is selected at random to write the vision and mission of the college. Let X and Y be the number of men and women on this committee, respectively. (a) Find the joint probability mass function of X and Y. (b) Find pX and pY, the marginal prob. mass functions of X and Y.

Joint distributions of 2 r.v.’s Sol:

Joint distributions of 2 r.v.’s Ex 8.2 Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. What is joint probability mass function? and what are pX and pY?

Joint distributions of 2 r.v.’s Sol: y pX(x) 1 2 3 4 5 6 x 1/12 1/6 1/24 2/24 1/48 3/48 1/96 4/96 6/96 1/192 5/192 10/192 1/384 6/384 15/384 20/384 pY(y) 63/384 120/384 99/384 64/384 29/384 8/384

Joint distributions of 2 r.v.’s Def Let A, B be the sets of possible values of X and Y, respectively.

Joint distributions of 2 r.v.’s Ex 8.3 Let the joint probability mass function of random variables X and Y be given by Find E(X) and E(Y).

Joint distributions of 2 r.v.’s

Joint distributions of 2 r.v.’s Theorem 8.1 Let p(x,y) be the joint prob. mass function of discrete random variables X and Y. If h is a function of two variables from R2 to R, then h(X,Y) is a diescrete random variable Coro E(X+Y)=E(X)+E(Y)

(Joint prob. Density functions) Def Two r. v.’s X and Y, defined on the same sample space, have a continuous joint distribution if there exists a nonnegative function of 2 variables, f(x,y) on RXR, such that for any region S in the xy-plane that can be formed from rectangles by a countable number of set operations, The function f(x,y) is called the joint probability density function of X and Y.

Joint distributions of 2 r.v.’s Def Let X and Y have joint density function f(x,y), then the functions are called, respectively, the marginal density functions of X and Y.

Joint distributions of 2 r.v.’s Def Let X and Y be two r. v.’s (discrete, continuous, or mixed). The joint distribution function of X and Y, is defined by

Joint distributions of 2 r.v.’s

Joint distributions of 2 r.v.’s Ex 8.4 The joint density function of random variables X and Y is given by (a) Determine the value of . (b) Find the marginal prob. density function of X and Y.

Joint distributions of 2 r.v.’s

Joint distributions of 2 r.v.’s Ex 8.5 Determine if F is the joint distribution function of 2 random variables X and Y.

Joint distributions of 2 r.v.’s

Joint distributions of 2 r.v.’s Ex 8.6 A circle of radius 1 is inscribed in a square with sides of length 2. A point is selected at random from the square we mean that the point is selected in a way that all the subsets of equal areas of the square are equally likely to contain the point.

Joint distributions of 2 r.v.’s

Joint distributions of 2 r.v.’s Def Let S be a subset of the plane with area A(S). A point is said to be randomly selected from S if for any subset R of S with area A(R), the probability that R contains the point is A(R)/A(S). Buffon’s needle problem(p.336) (an interesting geometric probability problem)

Joint distributions of 2 r.v.’s Ex 8.7 A man invites his fiancee to a fine hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 A.M. and 12 noon. If they arrive at random times during this period, what is the probability that they will meet within 10 minutes?

Joint distributions of 2 r.v.’s Sol:

Joint distributions of 2 r.v.’s Thm 8.2 Let f(x,y) be the density function of r. v. X and Y. If h is a function of two variables from R2 to R, then Z=h(X,Y) is a random variable with the expected value given by

Joint distributions of 2 r.v.’s Corollary For r. v.’s X and Y, E(X+Y) = EX + EY Proof:

Joint distributions of 2 r.v.’s Ex 8.9 Let X and Y have joint density function Find E(X2+Y2)

Joint distributions of 2 r.v.’s Sol:

8.2 Independent random variables Def Two random variables are called independent if, for A, B, in R, the events are independent, that is, if Using the axioms of probability, we can prove that X and Y are independent iff for any real numbers a and b,

Independent random variables Thm 8.3 Let X and Y be two random variables defined on the same sample space. If F is the joint distribution function of X and Y, then X and Y are independent iff for all real numbers t and u, F(t, u)=FX(t)FY(u).

Independent random variables Thm 8.4 Let X and Y be two discrete random variables defined on the same sample space. If p(x, y) is the joint probability mass function of X and Y, then X and Y are independent iff for all real numbers x and y, p(x, y)=pX(x)pY(y).

Independent random variables Thm 8.5 Let X and Y be independent random variables and g: RR and h: RR be real-valued functions; then g(X) and h(Y) are also independent random variables. Thm 8.6 Let X and Y be independent random variables and g: RR and h: RR be real-valued functions; then E[g(X)h(Y)]=E[g(X)]E[h(Y)]

Independent random variables By Thm 8.6, if X and Y are independent, then EXY=EXEY but the converse is not necessarily true. Ex 8.11 Let X be a r. v. with p(-1)=p(0)=p(1) =1/3. Letting Y=X2, we have EX=(-1+0+1)/3=0, EY=E(X2)=((-1)2+02+(1)2)/3=2/3, EXY=E(X3)=((-1)3+03+(1)3)/3=0. Thus EXY=EXEY while, clearly, X and Y are dependent.

Independent random variables Thm 8.7 Let X and Y be two continuous random variables defined on the same sample space. If f(x, y) is the joint density function of X and Y, then X and Y are independent iff for all real numbers x and y, f(x, y)=fX(x)fY(y).

Independent random variables Ex 8.12 Store A and B, which belong to the same owner, are located in 2 different towns. If the density function of the weekly profit of each store, in thousands of dollars, is given by and the profit of one store is independent of the other, what is the probability that next week one store makes at least $500 more than the other store?

Independent random variables Sol:

Independent random variables

Independent random variables Ex 8.15 Prove that 2 random variables X and Y with the following joint density function are not independent.

Independent random variables Sol:

8.3 Conditional distributions (Discrete case) Def Let X and Y be two discrete random variables. The conditional probability mass function of X given that Y=y is defined as follows:

Conditional distributions

Conditional distributions (Continuous case) Def Let X and Y be two continuous random variables with joint density f(x, y). The conditional density function of X given that Y=y is defined as follows:

Conditional distributions

Conditional distributions Ex 8.21 Let X and Y be continuous random variables with joint density function

Conditional distributions

Conditional distributions Ex 8.22 First, a point Y is selected at random from the interval (0, 1). Then another point X is chosen at random from (0, Y). Find the probability density function of X.

Conditional distributions

Conditional distributions Def The conditional expectation of the random variable X given that Y=y is (discrete) (continuous)

Conditional distributions Ex 8.24 Let X and Y be continuous random variables with joint density function

Conditional distributions

Skip 8.4 Transformations of 2 random vars