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Tails. Define a random variable X by X(Heads) = 1 and X(Tails) = 0.
The pmf of X is (approximately) the function f defined by
f(0) = P (X = 0) = .7,
f(1) = P (X = 1) = .3,
and f(x) = 0 for all x " X(S) = {0, 1}.
Example 3 A fair die is tossed and the number of dots on the upper
face is observed. The sample space is S = {1, 2, 3, 4, 5, 6}. Define a random
variable X by X(s) = 1 if s is a prime number and X(s) = 0 if s is not a
prime number.
The pmf of X is the function f defined by
f(0) = P (X = 0) = P ({4, 6}) = 1/3,
f(1) = P (X = 1) = P ({1, 2, 3, 5}) = 2/3,
and f(x) = 0 for all x " X(S) = {0, 1}.
Examples 1 3 have a common structure that we proceed to generalize.
Definition 3.3 A random variable X is a Bernoulli trial if X(S) = {0, 1}.
Traditionally, we call X = 1 a success and X = 0 a failure .
The family of probability distributions of Bernoulli trials is parametrized
(indexed) by a real number p " [0, 1], usually by setting p = P (X = 1).
We communicate that X is a Bernoulli trial with success probability p by
writing X
f defined by
f(0) = P (X = 0) = 1 - p,
f(1) = P (X = 1) = p,
and f(x) = 0 for all x " X(S) = {0, 1}.
Several important families of random variables can be derived from Ber-
noulli trials. Consider, for example, the familiar experiment of tossing a
fair coin twice and counting the number ofHeads. In Section 3.4, we will
generalize this experiment and count the number of successes in n Bernoulli
trials. This will lead to the family of binomial probability distributions.
58 CHAPTER 3. DISCRETE RANDOM VARIABLES
Bernoulli trials are also a fundamental ingredient of the St. Petersburg
Paradox, described in Example 7 of Section 3.3. In this experiment, a
fair coin is tossed untilHeadswas observed and the number ofTailswas
counted. More generally, consider an experiment in which a sequence of
independent Bernoulli trials, each with success probability p, is performed
until the first success is observed. Let X1, X2, X3, . . . denote the individual
Bernoulli trials and let Y denote the number of failures that precede the first
success. Then the possible values of Y are Y (S) = {0, 1, 2, . . .} and the pmf
of Y is
f(j) = P (Y = j) = P (X1 = 0, . . . , Xj = 0, Xj+1 = 1)
= P (X1 = 0) · · · P (Xj = 0) · P (Xj+1 = 1)
= (1 - p)jp
if j " Y (S) and f(j) = 0 if j " Y (S). This family of probability distributions
is also parametrized by a real number p " [0, 1]. It is called the geometric
family and a random variable with a geometric distribution is said to be a
geometric random variable, written Y
If Y
F (k) = P (Y d" k) = 1 - P (Y > k) = 1 - P (Y e" k + 1).
Because the event {Y e" k + 1} occurs if and only if X1 = · · · Xk+1 = 0, we
conclude that
F (k) = 1 - (1 - p)k+1.
Example 4 Gary is a college student who is determined to have a date
for an approaching formal. He believes that each woman he asks is twice
as likely to decline his invitation as to accept it, but he resolves to extend
invitations until one is accepted. However, each of his first ten invitations is
declined. Assuming that Gary s assumptions about his own desirability are
correct, what is the probability that he would encounter such a run of bad
luck?
Gary evidently believes that he can model his invitations as a sequence
of independent Bernoulli trials, each with success probability p = 1/3. If
so, then the number of unsuccessful invitations that he extends is a random
variable Y
10
2
.
P (Y e" 10) = 1 - P (Y d" 9) = 1 - F (9) = 1 - 1 - = .0173.
3
3.2. EXAMPLES 59
Either Gary is very unlucky or his assumptions are flawed. Perhaps
his probability model is correct, but p
the probability of success depends on who he asks. Or perhaps the trials
were not really independent.1 If Gary s invitations cannot be modelled as
independent and identically distributed Bernoulli trials, then the geometric
distribution cannot be used.
Another important family of random variables is often derived by con-
sidering an urn model. Imagine an urn that contains m red balls and n black
balls. The experiment of present interest involves selecting k balls from the
m+n
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