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# REVIEW OF PROBABILITY

This section contains a brief review of some definitions and results from probability that will be required in this book. Refer to a textbook on probability for more in-depth information, for example, Ghahramani (2004), Pitman (1993), Ross (2009), or Scheaffer and Young (2010).

## A.1 BASIC PROBABILITY

Recall that the set of all possible outcomes of a random experiment is called a sample space, S. An event E is a subset of S.

Proposition A.1 (Law of Total Probability) Let A denote an event in a sample space S and let B1, B2, . . . , Bn be a disjoint partition of A. Then

P(A) = P(B1)P(A|B1) + P(B2)P(A|B2) + · · · + P(Bn)P(A|Bn).

Definition A.1 A discrete random variable X is a function from S into the real numbers R with a range that is finite or countably infinite. That is, X: S → {x1, x2, . . . , xm}, or X: S → {x1, x2, . . .}.      ||

For instance, if we consider the experiment of rolling two dice, we can let X denote the sum of the two numbers that appear. Then X is a discrete random variable, X: S → {2, 3, . . . , 12}. The probability mass function (pmf) is a function p: R → [0, 1] such that p(x) = P(X = x), for all x in the range of X. Note then that Σx p(x) = 1, where the sum is over the range of X.

Definition A.2 A function X from S into the real numbers R is a continuous random variable if there exists a nonnegative function f such that for every subset C of R, P(X C) = f(x) dx. In particular, for ab, P(a < Xb) = f(x) dx.      || ...

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