Page 217 - Applied statistics and probability for engineers

Section 5-6/Moment-Generating Functions 195

5-95. A continuous random variable X has the following prob- 5-98. A random variable X has the gamma distribution
ability distribution: λ r −1 −λ x
(
f x) = 4 xe −2 x , x > 0 f x) = Γ r ( ) (λ x) e , x > 0
(
(a) Find the moment-generating function for X.
(a) Show that the moment-generating function of X is
(b) Find the mean and variance of X. r −
5-96 The continuous uniform random variable X has density ⎛ t ⎞
⎜
(
M t) = 1 − ⎟
X
function ⎝ λ ⎠
F x ( ) = 1 , α ≤ x ≤ β (b) Find the mean and variance of X.
β − α 5-99. Let X X 2 ,..., be independent exponential random
1 ,
X r
(a) Show that the moment-generating function is variables with parameter λ.
tβ
e − e tα (a) Find the moment-generating function of Y = X + X +
M t ( ) = 1 2
X
t( − α ) …+ X r.
β
(b) Use M X ( t) to ind the mean and variance of X. (b) What is the distribution of the random variable Y?
5-97. A random variable X has the exponential distribution 5-100. Suppose that X i has a normal distribution with mean μ i
2
(
f x) = λ e −λ x , x > 0 and variance σ i , i = 1, 2. Let X 1 and X 2 be independent.
(a) Find the moment-generating function of Y =
X +
X 1 .
1
Show that the moment-generating function of X is (b) What is the distribution of the random variable Y?
⎛ t ⎞
⎜
M t ( ) = 1 − λ ⎟ ⎠
X
⎝
(b) Find the mean and variance of X.
Supplemental Exercises
5-101. Show that the following function satisies the (e) Conditional mean of the number of people who suf-
properties of a joint probability mass function: fer severe side effects given that 19 suffer minor side
effects
x y f(x, y)
5-103. The backoff torque required to remove bolts in a steel
0 0 1 / 4
plate is rated as high, moderate, or low. Historically, the prob-
0 1 1 / 8
ability of a high, moderate, or low rating is 0.6, 0.3, or 0.1,
1 0 1 / 8 respectively. Suppose that 20 bolts are evaluated and that the
1 1 1 / 4 torque ratings are independent.
2 2 1 / 4 (a) What is the probability that 12, 6, and 2 bolts are rated as
Determine the following: high, moderate, and low, respectively?
(
(
.
(a) P X < 0 5 . ,Y <1 5) (b) P X ≤ ) 1 (b) What is the marginal distribution of the number of bolts
(
(c) P X <1 5. ) (d) P X > 0 5 ,Y <1 5) rated low?
(
.
.
(c) What is the expected number of bolts rated low?
(e) E(X), E(Y), V(X), V(Y).
(f) Marginal probability distribution of the random variable X (d) What is the probability that the number of bolts rated low
(g) Conditional probability distribution of Y given that X = 1 is more than two?
(h) E Y X| ( = ) 1 (e) What is the conditional distribution of the number of bolts
(i) Are X and Y independent? Why or why not? rated low given that 16 bolts are rated high?
(j) Correlation between X and Y. (f) What is the conditional expected number of bolts rated low
5-102. The percentage of people given an antirheumatoid given that 16 bolts are rated high?
medication who suffer severe, moderate, or minor side effects (g) Are the numbers of bolts rated high and low independent
are 10, 20, and 70%, respectively. Assume that people react random variables?
independently and that 20 people are given the medication. 5-104. To evaluate the technical support from a computer
Determine the following: manufacturer, the number of rings before a call is answered
(a) Probability that 2, 4, and 14 people will suffer severe, mod- by a service representative is tracked. Historically, 70% of the
erate, or minor side effects, respectively calls are answered in two rings or less, 25% are answered in
(b) Probability that no one will suffer severe side effects three or four rings, and the remaining calls require ive rings
(c) Mean and variance of the number of people who will suffer or more. Suppose that you call this manufacturer 10 times and
severe side effects assume that the calls are independent.
(d) Conditional probability distribution of the number of (a) What is the probability that eight calls are answered in two
people who suffer severe side effects given that 19 suffer rings or less, one call is answered in three or four rings, and
minor side effects one call requires ive rings or more?