Probability Mass Functions
• Definition: A probability mass function of a discrete random variable X is
pX (x) = P (X = x).
• Notation: For shorthand, we write
X ∼ pX
to say that a discrete random variable X has the distribution defined by the pmf pX (or that pX is the
pmf of X).
• Here are some simple examples of discrete random variables and their pmf’s.
1. Roll two fair 6-sided dice. Let X be the sum of the two independent rolls. For practice, you can
verify that

x−1

x = 2, 3, 4, 5, 6, 7
 36
14−x−1
pX (x) =
x = 8, 9, 10, 11, 12
36


0
otherwise.
2. Draw four cards from a standard deck without replacement. Let X be the number of hearts.
What is pX (x)?
Solution:
First, what is Im(X)?
Im(X) = {0, 1, 2, 3, 4}
We can use counting methods to find the pmf of X.
pX (x)
=
number of ways to get x hearts
.
number of ways to select 4 without replacement
For x = 0, 1, 2, 3, 4, this is
39 13
pX (x)
=
x
4−x
52
4
.
To summarize,
 13 39
 ( x )(4−x)
(52
pX (x) =
4)
0
otherwise.
3. Draw four cards from a standard deck with replacement. Let X be the number of hearts. What
is pX (x)? Convince yourself that the pmf of X can be written as in the table below:
x
pX (x)
0
.754
1
4(.25)(.75)2
2
6(.25)2 (.75)2
3
4(.25)3 (.75)2
4
.254
∗ This example has the same structure as the dartboard example from class.
∗ The four draws can be viewed as a sequence of iid Bernoulli trials with a success probability
of .25.
∗ X is the sum of four Bernoulli random variables: For i = 1, 2, 3, 4,
(
1 if card on ith draw is a heart
Xi =
0 otherwise.
∗ Then, X1 , X2 , X3 , and X4 are iid Bernoulli random variables with a success probability of
p = .25, and
X = X1 + X2 + X3 + X4 .
∗ X is called a binomial random variable with parameters n = 4 and p = .25.
∗ For practice, you can compute the expected value of X. Why does your answer make sense?
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