robability review RANDOM VARIABLES

A large chunk of probability is about random variables. Instead of giving a precise definition, let us just mention that a random variable can be thought of as an uncertain, numerical (i.e., with values in R) quantity. While it is true that we do not know with certainty what value a random variable X will take, we usually know how to assign a number the probability that its value will be in some some1 subset 1 We will not worry about measurability in this class. of R. For example, we might be interested in P[X ≥ 7], P[X ∈ [2, 3.1]] or P[X ∈ {1, 2, 3}]. The collection of all such probabilities is called the distribution of X. One has to be very careful not to confuse the random variable (a function on a probability space) itself and its distribution (a collection of probabilities, i.e., numbers). This point is particularly important when several random variables appear at the same time2. When two random variables X and Y have the same dis2 in addition to the fact that the knowing the distributions of X and Y does not determine the distribution of the pair (X, Y) or, even, X + Y. This is not the case with (nonrandom) numbers. If I know x and y, I know x + y. We’ll come back to this later. tribution, i.e., when P[X ∈ A] = P[Y ∈ A] for any set A, we say that