5.Joint Probability Distributions and Random Samples

1. Jointly Distribution Random Variable

The Joint Probability Mass Function for Two Discrete Random Variables

The joint pmf of two discrete rv's X and Y describes how much probability mass is placed on each posssible pair of values (x,y).

Let X and Y be two discrete rv's defined on the sample space δ of an experiment. The joint probability mass function p(x,y) is defined for each pair of numbers (x,y) by

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

Let A be any set consisting of pairs of (x,y) values. Then the probability P[(X,Y) →A]

The marginal probability mass functions of X and of Y, denoted by p_x(x) and p_Y(y), respectively, are given by

p_X(x) = ∑p(x,y)

p_Y(y) = ∑p(x,y)

The Joint Probability Density Function for Two Continuous Random Variables

Let X and Y be continuous rv's. Then ƒ(x,y) is the joint probability density function for X and Y if for any two-dimensional set A