Probability, Chapter 6: Jointly Distributed Random Variables

Joint Distribution Function (6.1)

이상준 교수 (덕성여대 수학과) 2015년 2학기

Textbook: Sheldon Ross, A first course in probability (9th ed, Pearson)

�� ����: ��, ���, �� (14�)

1

Motivation❖ We are often interested in considering two or more random variables.

❖ : joint distributed random variable

❖ Example:

❖ Tossing two dice: (X,Y) with X=1,2,…,6 and Y=1,2,…,6.

❖ For each person, we check (X,Y) with a height X and a weight Y.

Case Ⅰ: Discrete r. v.❖ Definition: Let X and Y be discrete random variables.

❖ � (X,Y) is the joint random variable of X and Y if its probability mass function is defined by p(x,y) = P{X=x,Y=y}.

❖ � The (marginal) probability mass function of X can be obtained by pX(x) = P{X=x} =

❖ Similarly, pY(y) = P{Y=y} =

3

❖ Example Ia: Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white, and 5 blue balls. If we let X and Y denote, respectively, the number of red and white balls chosen, then the joint probability mass function of X and Y, p(i,j) = P{X=i,Y=j}, is given by

4

(Reference: Ross, A first coursein probability, 9th ed)

❖ Example Ia: Suppose that 3 balls are randomly selected from an urn containing 3 red, 4 white, and 5 blue balls. If we let X and Y denote, respectively, the number of red and white balls chosen, then the joint probability mass function of X and Y, p(i,j) = P{X=i,Y=j}, is given by

5

(Reference: Ross, A first course in probability, 9th ed)

❖ Example Ib: Suppose that 15 percent of the families in a certain community have no children, 20 percent have 1 child, 35 percent have 2 children, and 30 percent have 3. Suppose further that in each family each child is equal likely (independently) to be a boy of a girl. If a family is chosen at random from this community, then B, the number of boys, and G, the number of girls, in this family will have the joint probability mass function shown in Table 6.2. What is the joint probability mass function for B and G?

6

(Reference: Ross, A first course in probability, 9th ed)

Case Ⅱ: Continuous r. v.❖ Definition: Let X and Y be continuous random variables

with density functions fX and fY.

❖ � (X,Y) is called a joint random variable of X and Y if there exists a function f(x,y) (x, y ∈ R) such that

❖ � f(x,y) is called the joint probability density function of (X,Y)

7

Density function of X and Y from (X,Y)❖ Fact: Let f(x,y) be the joint probability density function of (X,Y).

❖ � The probability density function of X is

❖ � The probability density function of Y is

❖ Definition: Let F(a,b) be the joint cumulative distribution function of (X,Y) if

❖ Property:

❖ �

❖ �

9

(Reference: Ross, A first course in probability, 9th ed)

❖ Example Ic: The joint density function of X and Y is given by

❖ Compute (a) P{X>1, Y<1}, (b) P{X<Y}, and (c) P{X<a}.

❖ Solution: (a)

10

(Reference: Ross, A first course in probability, 9th ed)

❖ (b)

❖ (c)

11

❖ Solution: (by lecturer)

12

Joint r.v. with n r.v.❖ Similarly, we can also define joint probability distribution for n random variables.

❖ Definition:

1. (X1,X2,…,Xn) is called a joint random variable of X1,X2,…,Xn if there exists a function f(x1,x2,…,xn) such that, for any set C in Rn,

2. f(x1,x2,…,xn) is the joint density function of (X1,X2,…,Xn).

3. The joint distribution function F(a1,a2,…,an) is defined by F(a1,a2,…,an) = P{X1≤a1,X2,≤a2,…,Xn≤an}.

❖ Example If: (The multinomial distribution) One of the most important joint distribution is the multinomial distribution, which arises when a sequence of n independent and identical experiments is performed. Suppose that each experiment can result in any one of r possible outcomes, with respective probabilities If we let Xi, denote the number of the n experiments that result in out come number i, then whenever

14