Joint Distribution Functions/ part1

FX,Y( x, y)
Conditional Distributions
The joint distribution of random variables X and Y use
the pdf of the joint distribution, denoted ( , ). , f x y X Y This pdf is usually given, although some problems only give it up to a constant. The methods for solving problems involving joint distributions are similar to the methods for single random variables, except that we work with double integrals and 2-dimensional probability spaces instead of single integrals and 1-dimensional probability space . We illustrate these methods by example.
Discrete Case: Analogous to the discrete single random variable case, we
have
Pr ((X = x) ? (Y = y, )) ? 1
The Mixed Case (one of the random variables is discrete, the other is
continuous) is also illustrated with examples.
cdf of Joint Distribution – denoted FX,Y( x, y)
Then the marginal pdf s (or pmf s
= probability mass functions, if you prefer this terminology for discrete random
variables) are defined by
fY(y) = P(Y = y) and fX(x) = P(X = x).
The joint pdf is, similarly,
fX,Y(x,y) = P(X = x and Y = y).
The conditional pdf of the conditional distribution Y|X is
fY|X(y|x) = P(Y = y|X = x)