Lectures on Elementary Probability

Chapter 3
Discrete Random Variables
3.1 Mean
A discrete random variable X is a function from the sample space S that has
a finite or countable infinite number of real numerical values. For a discrete
random variable each event X = x has a probability P [X = x]. This function
of x is called the probability mass function of the random variable X.
The mean or expectation of X is
µ X = E[X] = ∑ x
xP [X = x], (3.1)
where the sum is over the values of X.
One special case of a discrete random variable is a random variable whose
values are natural numbers. Sometimes for technical purposes the following
theorem is useful. It expresses the expectation in terms of a sum of probabilities.
Theorem 3.1 Let Y be a random variable whose values are natural numbers.
Then
∞∑
E[Y ] = P [Y ≥ j]. (3.2)
Proof: We have
∞∑
∞∑
E[Y ] = kP [Y = k] =
k=1
On the other hand,
j=1
k=1 j=1
k∑
∞∑ ∞∑
P [Y = k] = P [Y = k]. (3.3)
j=1 k=j
∞∑
P [Y = k] = P [Y ≥ j]. (3.4)
k=j
Note that if X is a discrete random variable, and g is a function defined for
the values of X and with real values, then g(X) is also a random variable.
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