Random Variables If  is an outcome space with a probability measure and X is a real-valued function defined over the elements of , then X is a random variable. Standard notation –Capital letter for a random variable (e.g., X) –Lower-case letter for a realization of the random variable (e.g., x)

Example Flip a coin until the first tail or until the 4 th flip, whichever comes first. Let X represent the number of heads observed. –What’s the range of X? –What’s the probability distribution of X?

Joint Distributions Given two random variables X and Y defined for in the same setting, we can consider the joint outcome (X, Y) as a random pair of values. –The event (X = x, Y = y) is the intersection of the events (X = x) and (Y = y). –The distribution of (X, Y) is called the joint distribution of X and Y.

Marginal Probabilities & Distributions Given a joint distribution of X & Y: –The marginal probability that X = x is –The distribution of X (irrespective of Y) is called the marginal distribution of X. As x varies over the range of X, the marginal probabilities that X = x define the marginal distribution of X.

Conditional Distributions Given X = x, as y varies over the range of Y the probabilities P(Y=y|X=x) define a probability distribution over the range of Y. This distribution is called the conditional distribution of Y given X = x.