AMS 507, Lecture 7
Chapter Four: Random Variables
4.1. Random Variables
Definition: Let S be the sample space of an experiment. A real-valued function
X : S  → R is called a random variable of the experiment if, for each interval
I ⊆ R, {s: X ( s) ∈ I } is an event.
Definition: If X is a random variable, then the function F defined on ( − ∞ , ∞ ) by
F (t ) = P( X ≤ t ) is called the cumulative distribution function of X.
Some authors use the term distribution function rather than cumulative distribution
function (cdf ).
4.2. Discrete Random Variables
For a discrete random variable X, we define the probability mass function (pmf) p(a) of
X by p (a ) = P{ X = a}.
Example 2a. The probability mass function of a random variable X is given by
cλi
p (i ) =
, i = 0,1, 2,  , where λ > 0. Find P{ X = 0} and P{ X > 2}.
i!
4.3. Expected Value
Definition The expected value of a discrete random variable X with the probability
function p(x) and set of possible values A (that is, those values x with p(x)>0) is defined
by
E ( X ) = ∑ xp( x ).
x∈A
We say that E(X) exists if this sum converges absolutely.
If X is a constant random variable, that is, if P( X = c) = 1 for a constant c, then
E ( X ) = c.
Example: The random variable X takes the value 0 with probability 1-p and the value 1
with probability p, 0<p<1. Find E(X).
4.4. Expectation of a Function of a Random Variable
Proposition 4.1. Let X be a discrete random variable with set of possible values A and
probability function p(x), and let g be a real-valued function. Then g(X) is a random
variable with
E[ g ( X )] = ∑ g ( x ) p( x ).
x∈A
Corollary 4.1. If a and b are constants, then E (aX + b) = aE ( X ) + b.
Corollary Let X be a discrete random variable; g1 , g 2 , , g n be real-valued functions,
and let α 1 , α 2 , , α n be real numbers. Then
E[α 1 g1 ( X ) + α 2 g 2 ( X ) + + α n g n ( X )] = α 1 E[ g1 ( X )] + α 2 E[ g 2 ( X )]+ + α n E[ g n ( X )].
4.5. Variance
Definition Let X be a discrete random variable with a set of possible values A, probability
mass function p(x), and E ( X ) = µ . Then σ X and Var(X), called the standard deviation
and the variance of X, respectively, are defined by
var( X ) = E[( X − µ ) 2 ] and σ X = var( X ).
Exceptionally useful identify: var( X ) = E ( X 2 ) − [ E ( X )]2 .
Example: Let X be distributed as a Poisson random variable with mean λ. What is the
variance of X?
Var(X)=0 if and only if X is a constant with probability 1.
var(aX + b) = a 2 var( X ).
σ aX + b = | a|σ X .
End of handout