Conditional Expectation

If X and Y are discrete, this distribution can be described with a joint probability mass function. If X and Y are continuous, this distribution can be described with a joint probability density function.
If we are given a joint probability distribution for X and Y , we can obtain the individual probability distribution for X or for Y (and these are called the Marginal Probability Distributions.
Because the the probability mass functions for X and Y appear in the margins of the table i.e. column and row totals), they are often referred to as the Marginal Distributions for X and Y .
When there are two random variables of inter-
est, we also use the term bivariate probability distribution or bivariate distribution to refer to the joint distribution. Marginal Probability Mass Function
If X and Y are discrete random variables with joint probability mass function fXY (x, y)
Suppose that X and Y are discrete random variables. If the events X ? x and Y ? y are independent events for all
x and y, then we say that X and Y are independent random variables. In such case,
(27)
or equivalently
f (x, y) ? f1(x)f2(y) (28)
Conversely, if for all x and y the joint probability function f(x, y) can be expressed as the product of a function
of x alone and a function of y alone (which are then the marginal probability functions of X and Y), X and Y are
independent. If, however, f(x, y) cannot be so expressed, then X and Y are dependent.
If X and Y are continuous random variables, we say that they are independent random variables if the events
X ? x and Y ? y are independent events for all x and y. In such case we can write
P(X ? x, Y ? y) ? P(X ? x)P(Y ? y) (29)
or equivalently
F(x, y) ? F1(x)F2(y) (30)
where F1(z) and F2(y) are the (marginal) distribution functions of X and Y, respectively. Conversely, X and Y are
independent random variables if for all x and y, their joint distribution function F(x, y) can be expressed as a prod-
uct of a function of x alone and a function of y alone (which are the marginal distributions of X and Y, respec-
tively). If, however, F(x, y) cannot be so expressed, then X and Y are dependent.
For continuous independent random variables, it is also true that the joint density function f(x, y) is the prod-
uct of a function of x alone, f1(x), and a function of y alone, f2(y), and these are the (marginal) density functions
of X and Y, respectively.
Change of Variables
Given the probability distributions of one or more random variables, we are often interested in finding distribu-
tions of other random variables that depend on them in some specified manner. Procedures for obtaining these
distributions are presented in the following theorems for the case of discrete and continuous variables.
P(X ? x, Y ? y) ? P(X ? x)P(Y ? y)