Math 461 Introduction to Probability A.J. Hildebrand
Continuous Random Variables
•Definition:
A random variable Xis called continuous if it satisfies P(X=x) = 0 for each x.1Informally, this
means that Xassumes a “continuum” of values. By contrast, a discrete random variable is one that
has a finite or countably infinite set of possible values xwith P(X=x)>0 for each of these values.
•Cumulative distribution function (c.d.f.):
– Definition:
F(x) = P(X≤x).
– Properties:
F(x) is non-decreasing,
lim
x→−∞ F(x) = 0,lim
x→∞ F(x) = 1.
•Probability density function (p.d.f.):
– Definition:
f(x) = F0(x),
– Properties:
f(x)≥0,Z∞
−∞
f(x)dx = 1.
•Probability computations via p.d.f.’s and c.d.f.’s:
P(a≤X≤b) = Zb
a
f(x)dx =F(b)−F(a)
•Expectation and variance: The definitions and properties for expectation and variance are analogous
to those for discrete random variables, with sums replaced by integrals:
E(X) = Z∞
−∞
xf(x)dx, Var(X) = E(X2)−E(X)2,
E(cX) = cE (X), E(X+Y) = E(X) + E(Y),
E(g(X)) = Z∞
−∞
g(x)f(x)dx.
1This is the standard definition of a continuous random variable. The Ross text has a more narrow definition of a continuous
random variable, but the difference between these two definitions is immaterial for this course.
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