Joint Probability Distributions: Understanding the Relationship between Random Variables, Study notes of Statistics

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STAT 430/510 Probability

Hui Nie

Lecture 13

June 18th, 2009

Introduction

Many random variables are naturally related to each other. Forecast and actual weather Height and weight Lifetimes of a system and a component These random variables are usually studied together instead of individually The joint behavior of several random variables is described by their joint distribution.

Joint Distribution: Continued

All joint probability statements about X and Y can be answered in terms of their joint distribution function. Example: P(x 1 < X ≤ x 2 , y 1 < y ≤ y 2 ) = F (x 2 , y 2 )+F (x 1 , y 1 )−F (x 2 , y 1 )−F (x 1 , y 2 )

Joint pmf of Discrete Random Variable

When X and Y are two discrete random variables, the joint pmf p(x, y) is defined for each pair of numbers (x,y) by

p(x, y) = P(X = x, Y = y)

Let A be any set consisting of pairs of (x,y) values. Then

P[(X , Y ) ∈ A] =

(x,y)∈A

p(x, y)

Example

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 equally likely (independently) to be a boy or 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 as below.

0 1 2 3 Row sum= P(B = i) 0 .15 .10 .0875 .0375 0. 1 .10 .175 .1125 0 0. 2 0.0875 0.1125 0 0 0. 3 0.0375 0 0 0 0. Column sum= P(G = j) 0.3750 0.3875 0.2000 0.

Joint pdf of Continuous Random Variable

X and Y are two continuous r.v.’s. Then f (x, y) is the joint pdf for X and Y if for any 2-dimensional set A,

P[(X , Y ) ∈ A] =

(x,y)∈A

f (x, y)dxdy

Marginal pdf of Continuous Random Variable

The marginal pdf’s of X and Y , denoted by fX (x) and fY (y), respectively, are given by

fX (x) =

−∞

f (x, y)dy, for − ∞ < x < ∞

fY (y) =

−∞

f (x, y)dx, for − ∞ < y < ∞

The marginal cdf’s of X and Y, denoted by FX and FY , are given by

FX (x) = P(X ≤ x) = P(X ≤ x, Y < ∞) = F (x, ∞) FY (y) = P(Y ≤ y) = P(X < ∞, Y ≤ y) = F (∞, y)

Example

The joint density function of X and Y is given by

f (x, y) =

2 e−x^ e−^2 y^ , 0 < x < ∞, 0 < y < ∞ 0 , otherwise Compute (a) P(X > 1 , Y < 1 ) (b) P(X < Y ) (c) The marginal density of X

Joint Distribution

The joint distribution of n r.v.’s X 1 , · · · , Xn can be defined in the same manner as n = 2. The joint cumulative probability distribution function F (x 1 , · · · , xn) is defined as F (x 1 , · · · , xn) = P(X 1 ≤ x 1 , · · · , Xn ≤ xn)

Joint Distribution

If X 1 , X 2 , · · · , Xn are discrete r.v.’s, the joint pmf is p(x 1 , · · · , xn) = P(X 1 = x 1 , · · · , Xn = xn) n r.v.’s X 1 , X 2 , · · · , Xn are said to be jointly continuous if there exists a function f (x 1 , · · · , xn) such that for all measurable set A in Rn P((X 1 , · · · , Xn) ∈ A) =

(x 1 ,··· ,xn)∈A f^ (x^1 ,^ · · ·^ ,^ xn)dx^1 · · ·^ dxn