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2. Expectations of Functions of Random Variables 89

One should verify that

where P (t) is the pgf of X defined in (2.5.1).
X

2.6 Exercises and Complements

2.2.1 A game consists of selecting a three-digit number. If one guesses the
number correctly, the individual is paid $800 for each dollar the person bets.
Each day there is a new winning number. Suppose that an individual bets $1
each day for a year, then how much money can this individual be expected to
win?
2.2.2 An insurance company sells a life insurance policy with the face value
of $2000 and a yearly premium of $30. If 0.4% of the policy holders can be
expected to die in the course of a year, what would be the company’s expected
earnings per policyholder in any year?
2.2.3 (Exercise 1.5.1 Continued) Suppose that a random variable X has the
following pmf:

X values: –2 0 1 3 8

Probabilities: .2 p .1 2p .4
where p ∈(0, .1]. Evaluate µ and σ for the random variable X.
2.2.4 (Exercise 1.5.2 Continued) Suppose that a random variable X has the
following pmf:
X values : –2 0 3 5 8
Probabilities: .2 p .1 .3 – p .4
where p ∈ (0, .3). With a fixed but arbitrary p ∈ (0, .3), find the expressions
of µ and σ for the random variable X depending upon p. Is there some p ∈ (0,
.3) for which the V(X) is minimized?
2.2.5 Consider a random variable X which has the following discrete uni-
form distribution along the lines of (1.7.11):

Derive the explicit expressions of µ and σ for this distribution. {Hint: Use the
first two expressions from (1.6.11).}