##### Document Text Contents

Page 1

Introduction to Probability

SECOND EDITION

Dimitri P. Bertsekas and John N. Tsitsiklis

Massachusetts Institute of Technology

WWW site for book information and orders

http://www.athenasc.com

~ Athena Scientific, Belmont, Massachusetts

Page 269

Problems

(b) We have

b

L 1 sk - e b - a + l k=a

sa b-a

_

e � esk b - a + l L..,. k=O

esa es(b-a+l ) - 1

b - a + l

sx Jb eSx eSb - esa M(s) = E[e 1 = a b _ a dx = s(b - a) .

259

Problem 40. * Suppose that the transform associated with a discrete random variable

X has the form

A(eS) M(s) = B(es ) '

where A(t) and B(t) are polynomials of the generic variable t . Assume that A(t) and

B(t) have no common roots and that the degree of A(t) is smaller than the degree of

B(t) . Assume also that B(t) has distinct, real, and nonzero roots that have absolute

value greater than 1 . Then it can be seen that M(s) can be written in the form

where I/Tl , . . . , I/Tm are the roots of B(t) and the ai are constants that are equal to

limes_ .!. ( 1 - TieS)M(s) , i = 1 , . . . , m. ri

(a) Show that the PMF of X has the form

P(X = k) = {f aiTf ' t= l

0,

if k = 0, 1 , . . . ,

otherwise.

Note: For large k, the PMF of X can be approximated by a-;TI ' where I is the

index corresponding to the largest ITi l (assuming I is unique) .

(b) Extend the result of part (a) to the case where M(s) = ebs A(eS ) / B (eS ) and b is

an integer.

Solution. (a) We have for all s such that ITi les < 1

Therefore,

Page 270

260 Further Topics on Random Variables

and by inverting this transform, we see that

m

P(X = k) I: air:

t=l

Chap. 4

for k ;::: 0, and P(X = k) = 0 for k < O. Note that if the coefficients a, are nonnegative,

this PMF is a mixture of geometric PMFs.

(b) In this case, AI (s) corresponds to the translation by b of a random variable whose

transform is A(e8 )/ B(e8 ) (cf. Example 4 .25 ) , so we have

P(X = k) = { �; a,rj'-b) , if k = b, b + 1 " " ,

0 , otherwise.

SECTION 4.4. Sum of a Random Number of Independent Random

Variables

Problem 41. At a certain time, the number of people that enter an elevator is a

Poisson random variable with parameter A. The weight of each person is independent

of every other person's weight, and is uniformly distributed between 100 and 200 lbs.

Let Xi be the fraction of 100 by which the ith person exceeds 100 lbs, e.g. , if the 7th

person weighs 175 lbs. , then X7 0.75. Let Y be the sum of the Xi .

(a) Find the transform associated with Y.

(b) Use the transform to compute the expected value of Y.

(c) Verify your answer to part (b) by using the law of iterated expectations.

Problem 42. Construct an example to show that the sum of a random number of

independent normal random variables is not normal (even though a fixed sum is) .

Problem 43. A motorist goes through 4 lights, each of which is found to be red with

probability 1/2 . The waiting times at each light are modeled as independent normal

random variables with mean 1 minute and standard deviation 1 /2 minute. Let X be

the total waiting time at the red lights.

(a) Use the total probability theorem to find the PDF and the transform associated

with X, and the probability that X exceeds 4 minutes. Is X normal?

(b) Find the transform associated with X by viewing X as a sum of a random number

of random variables.

Problem 44. Consider the calculation of the mean and variance of a sum

Y = X1 + " , + XN ,

where N is itself a sum of integer-valued random variables, i .e. ,

Page 539

ATHENA SCIENTIFIC BOOKS

OPTIMIZATION AND COMPUTATION SERIES

Introduction to Probability

SECOND EDITION

Dimitri P. Bertsekas and John N. Tsitsiklis

Massachusetts Institute of Technology

WWW site for book information and orders

http://www.athenasc.com

~ Athena Scientific, Belmont, Massachusetts

Page 269

Problems

(b) We have

b

L 1 sk - e b - a + l k=a

sa b-a

_

e � esk b - a + l L..,. k=O

esa es(b-a+l ) - 1

b - a + l

sx Jb eSx eSb - esa M(s) = E[e 1 = a b _ a dx = s(b - a) .

259

Problem 40. * Suppose that the transform associated with a discrete random variable

X has the form

A(eS) M(s) = B(es ) '

where A(t) and B(t) are polynomials of the generic variable t . Assume that A(t) and

B(t) have no common roots and that the degree of A(t) is smaller than the degree of

B(t) . Assume also that B(t) has distinct, real, and nonzero roots that have absolute

value greater than 1 . Then it can be seen that M(s) can be written in the form

where I/Tl , . . . , I/Tm are the roots of B(t) and the ai are constants that are equal to

limes_ .!. ( 1 - TieS)M(s) , i = 1 , . . . , m. ri

(a) Show that the PMF of X has the form

P(X = k) = {f aiTf ' t= l

0,

if k = 0, 1 , . . . ,

otherwise.

Note: For large k, the PMF of X can be approximated by a-;TI ' where I is the

index corresponding to the largest ITi l (assuming I is unique) .

(b) Extend the result of part (a) to the case where M(s) = ebs A(eS ) / B (eS ) and b is

an integer.

Solution. (a) We have for all s such that ITi les < 1

Therefore,

Page 270

260 Further Topics on Random Variables

and by inverting this transform, we see that

m

P(X = k) I: air:

t=l

Chap. 4

for k ;::: 0, and P(X = k) = 0 for k < O. Note that if the coefficients a, are nonnegative,

this PMF is a mixture of geometric PMFs.

(b) In this case, AI (s) corresponds to the translation by b of a random variable whose

transform is A(e8 )/ B(e8 ) (cf. Example 4 .25 ) , so we have

P(X = k) = { �; a,rj'-b) , if k = b, b + 1 " " ,

0 , otherwise.

SECTION 4.4. Sum of a Random Number of Independent Random

Variables

Problem 41. At a certain time, the number of people that enter an elevator is a

Poisson random variable with parameter A. The weight of each person is independent

of every other person's weight, and is uniformly distributed between 100 and 200 lbs.

Let Xi be the fraction of 100 by which the ith person exceeds 100 lbs, e.g. , if the 7th

person weighs 175 lbs. , then X7 0.75. Let Y be the sum of the Xi .

(a) Find the transform associated with Y.

(b) Use the transform to compute the expected value of Y.

(c) Verify your answer to part (b) by using the law of iterated expectations.

Problem 42. Construct an example to show that the sum of a random number of

independent normal random variables is not normal (even though a fixed sum is) .

Problem 43. A motorist goes through 4 lights, each of which is found to be red with

probability 1/2 . The waiting times at each light are modeled as independent normal

random variables with mean 1 minute and standard deviation 1 /2 minute. Let X be

the total waiting time at the red lights.

(a) Use the total probability theorem to find the PDF and the transform associated

with X, and the probability that X exceeds 4 minutes. Is X normal?

(b) Find the transform associated with X by viewing X as a sum of a random number

of random variables.

Problem 44. Consider the calculation of the mean and variance of a sum

Y = X1 + " , + XN ,

where N is itself a sum of integer-valued random variables, i .e. ,

Page 539

ATHENA SCIENTIFIC BOOKS

OPTIMIZATION AND COMPUTATION SERIES