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370 Appendix A Random Variables and Probability Distributions
where {x0,x1,x2,...} is a finite or countably infinite set. In the case (A.1.2) we shall
say that the random variable X is continuous. The function f is called the probability
density function (pdf) of X and can be found from the relation
f(x) F ′ (x).
In case (A.1.3), the possible values of X are restricted to the set {x0,x1,...}, and
we shall say that the random variable X is discrete. The function p is called the
probability mass function (pmf) of X, and F is constant except for upward jumps
of size p(xj) at the points xj. Thus p(xj) is the size of the jump in F at xj, i.e.,
p(xj) F(xj) − F(x −
j ) P [X xj],
where F(x −
j ) limy↑xj F(y).
Examples of Continuous Distributions
(a) The normal distribution with mean µ and variance σ 2 . We say that a random
variable X has the normal distribution with mean µ and variance σ 2 written
more concisely as X ∼ N µ, σ 2 if X has the pdf given by
n x; µ, σ 2 (2π) −1/2 σ −1 e −(x−µ)2 /(2σ 2 ) −∞