Page 115 - Probability and Statistical Inference

92 2. Expectations of Functions of Random Variables

2.2.19 Prove Theorem 2.2.4. {Hint: By the Theorem 2.2.1, note that E(aX
+ b) = aE(X) + b. Next, apply the Definition 2.2.2 and write V(aX + b) =
E{(aX + b) – [aE(X) + b]} = E{[aX – aE(X)] } = a E{[X – E(X)] } which
2
2
2
2
simplifies to a V(X).}
2
2.2.20 Suppose that X is a random variable whose second moment is
finite. Let g(a) = E[(X – a) ], a ∈ ℜ. Show that g(a) is minimized with respect
2
to a when a = µ, the mean of X. {Hint: Write out E[(X – a) ] = E[X ] –
2
2
2aE[X] + a . Treat this as a function of a single variable a and minimize this
2
with respect to a. Refer to (1.6.27).}
2.2.21 In the Theorem 2.2.5, we had assumed that
for the df F(x) of a non-negative continuous random variable X. Write x{1 –
–1
F(x)} = {1 – F(x)}/x and then it is clear that unfortu-
nately takes the form of 0/0. Hence, by applying the L’Hôpital’s rule (refer to
(1.6.29)) show that we may assume instead where f(x) is
the associated pdf.
2.2.22 (Exercise 2.2.21 Continued) Suppose that X has the exponential
–1 –x/β
distribution with its pdf f(x) = β e for x > 0, β > 0. Obtain the expression
of µ by applying the integral expression found in the Theorem 2.2.5. Is the
sufficient condition satisfied here? How about the suffi-
cient condition ?
2.2.23 This exercise provides an application of the Theorem 2.2.6. Sup-
pose that X has the Geometric(p) distribution, defined in (1.7.7), with its pmf
x–1
f(x) = pq , x = 1, 2, ..., 0 < p < 1, q = 1 – p.
x
(i) Show that F(x) = 1 – q , x = 1, 2, ...;
(ii) Perform the infinite summation using the ex-
pression for , and hence find µ
A direct approach using the Definition 2.2.1 to find µ was sought out in
the Exercise 2.2.6.
2.2.24 (Monotone Convergence Theorem) Let {X ; n = 1} be a se-
n
quence of non-negative random variables. Then, show that

{Note: We state this result here for completeness. It has been utilized in the
proof of the Theorem 2.2.6. It may be hard to prove this result at the level of
this book. See also the Exercise 5.2.22 (a).}
2.3.1 Suppose that Z has the standard normal distribution. By directly
evaluating the appropriate integrals along the lines of (2.2.22)-(2.2.24) and
Example 2.3.4 show that