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Reserve Supplemental Exercises Chapter 5 Problem 6

To evaluate the technical support from a computer manufacturer, the number of rings before a

call is answered by a service representative is tracked. Historically, 70% of the calls are

answered in two rings or less, 25% are answered in three or four rings, and the remaining calls

require five rings or more. Suppose that you call this manufacturer 10 times and assume that the

calls are independent. Let X, Y, and Z denote the number of calls answered in two rings or less, in

three or four rings, and in five rings or more, respectively.

(a) What is the probability that 7 calls are answered in two rings or less, 2 cells are answered in

three or four rings, and one call requires five rings or more?

(b) Let W denote the number of calls answered in four rings or less. What is the probability that

all 10 calls are answered in four rings or less?

(c) What is the expected number of calls answered in four rings or less?

(d) What is the conditional distribution of the number of calls requiring five rings or more given

that eight calls are answered in two rings or less?

(e) What is the conditional expected number of calls requiring five rings or more given that eight

calls are answered in two rings or less?

(f) Are the number of calls answered in two rings or less and the number of calls requiring five

rings or more independent random variables?

SOLUTION

(a)

(e)

Reserve Supplemental Exercises Chapter 5 Problem 7

The length and width of panels used for interior doors (in inches) are denoted as X and Y,

respectively. Suppose that X and Y are independent, continuous uniform random variables for

17.75 18.25x

and

4.75 5.25y

, respectively.

(a) By integrating the joint probability density function over the appropriate region, determine

the probability that the area of a panel exceeds 93 square inches.

(b) What is the probability that the perimeter of a panel exceeds 46 inches?

SOLUTION

(a)

(b)

Reserve Supplemental Exercises Chapter 5 Problem 8

If determine

( )

EX

,

( )

EY

,

( )

VX

,

( )

VY

, and

by reorganizing the parameters in the joint

probability density function.

( )

EX

=

( )

EY

=

( )

VX

=

( )

VY

=

=

SOLUTION

Reserve Supplemental Exercises Chapter 5 Problem 9

Use moment generating functions to determine the normalized power

( )

4 1/4

[]EX

from a cycling

power meter when X has a normal distribution with mean 210 and standard deviation 20 Watts.

SOLUTION

The moment generating function for a normal distribution is

Reserve Supplemental Exercises Chapter 5 Problem 10

Show that if

1

X

,

2

X

, ...,

p

X

are independent, continuous random variables,

( )

( ) ( )

( )

1 1 2 2 1 1 2 2

, ,..., ...

p p p p

P X A X A X A P X A P X A P X A =

for any regions

1

A

,

2

A

,

...,

p

A

in the range of

1

X

,

2

X

, ...,

p

X

, respectively. To do so, complete the following

derivation by selecting the correct answers.

By the ______,

( ) ( )

1 2 1 2

1 1 2 2 ... 1 2 1 2

, ,..., ... , ,..., d d ...d

pp

p p A A A X X X p p

P X A X A X A f x x x x x x =

From the _______,

( )

( ) ( )

( )

1 2 1 2

... 1 2 1 2

, ,..., ...

pp

X X X p X X X p

f x x x f x f x f x=

Therefore,

( )

( ) ( )

( )

1 2 1 2 1 2

12

... 1 2 1 2 1 1 2 2

... , ,..., d d ...d d d ... d

p p p

p

A A A X X X p p X X X p p

A A A

f x x x x x x f x x f x x f x x

=

By the _______,

( ) ( )

( )

( ) ( )

( )

12

12

1 1 2 2 1 1 2 2

d d ... d ...

p

p

X X X p p p p

A A A

f x x f x x f x x P X A P X A P X A

=

SOLUTION

By the definition,

Reserve Supplemental Exercises Chapter 5 Problem 11

Show that if

1

X

,

2

X

, ...,

p

X

are independent random variables and

1 1 2 2 ... pp

Y c X c X c X= + + +

, then

( ) ( ) ( )

( )

2 2 2

1 1 2 2 ... pp

V Y c V X c V X c V X= + + +

.

SOLUTION

By the definition,

Reserve Supplemental Exercises Chapter 5 Problem 12

Suppose that the joint probability function of the continuous random variables X and Y is

constant on the rectangle

0xa

,

0yb

. Show that X and Y are independent.

Express your answers in terms of a and b. Determine the constant c such that

( )

,

XY

f x y c=

for

0xa

,

0yb

.

SOLUTION

Reserve Supplemental Exercises Chapter 5 Problem 13

Suppose that the range of the continuous variables X and Y is

0xa

and

0yb

. Also

suppose that the joint probability density function

( ) ( ) ( )

,

XY

f x y g x h y=

where

( )

gx

is a

function only of x, and

( )

hy

is a function only of y. Show that X and Y are independent.

SOLUTION

The marginal probability density function of X is

Reserve Supplemental Exercises Chapter 5 Problem 14

This exercise extends the hypergeometric distribution to multiple variables. Consider a

population with N items of k different types. Assume that there are

1

N

items of type 1,

2

N

items of type 2, …,

k

N

items of type k so that

12

... k

N N N N+ + + =

. Suppose that a random

sample of size n is selected, without replacement, from the population. Let

1

X

,

2

X

, …,

k

X

denote the number of items of each type in the sample so that

12

... n

X X X n+ + + =

. Show that

for feasible values of n,

1

x

,

2

x

, …,

k

x

,

1

N

,

2

N

, …,

k

N

, the probability is

( )

12

12

1 1 2 2

...

, ,...,

k

k

kk

N

NN

x

xx

P X x X x X x N

n

= = = =

.

SOLUTION

Reserve Supplemental Exercises Chapter 5 Problem 15

Use the properties of moment-generating functions to show that a sum of P independent normal

random variables with means

i

and variances

2

i

for

1, 2,...,ip=

has a normal distribution.

Find mean and variance for this distribution.

The moment generating function of a normal random variable with mean

and standard

deviation

is ____.

Mean

of the sum of p independent normal random variables is equal to _____.

Variance

2

of the sum of p independent normal random variables is equal to _____.

SOLUTION

Reserve Supplemental Exercises Chapter 5 Problem 16

Show that by expanding

tX

e

in a power series and taking expectations term by term we may

write the moment-generating function as

( )

( )

2

12

1 ... ...

2! !

r

tX s s s

Xr

tt

M t E e t r

= = + + + + +

. Thus,

the coefficient of

!

r

t

r

in this expansion is

s

r

, the rth origin moment. Use the power series

expansion for

( )

X

Mt

for a gamma random variable to determine

1

s

and

2

s

.

SOLUTION

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