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).

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 B*_{1}*, B*_{2}*, . . . , B _{n} be a disjoint partition of A. Then*

*P*(*A*) = *P*(*B*_{1})*P*(*A*|*B*_{1}) + *P*(*B*_{2})*P*(*A*|*B*_{2}) + · · · + *P*(*B _{n}*)

**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* → {*x*_{1}, *x*_{2}, . . . , *x _{m}*}, or

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 Σ_{x} *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 *a* ≤ *b*, *P*(*a < X* ≤ *b*) = *f*(*x*) *dx*. || ...

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