Fundamentals of Probability and Statistics for Engineers

times equally likely to choose either direction, determine the average time interval

(in minutes) that the miner will be trapped.

4.15 Show that:

4.16 Let random variable X be uniformly distributed over interval 0 Deter-
mine a lower bound for using the Chebyshev inequality and
compare it with the exact value of this probability.

4.17 For random variable X defined in Problem 4.16, plot as a func-
tion of h and co mpare it with its lo wer bound as determined by the Chebyshev
inequality. Show that the lower bound becomes a better approximation of
becomes large.

4.18 Let a random variable X take only nonnegative values; show that, for any a > 0,

This is known as Markov’s inequality.

4.19 The yearly snowfall of a given region is a random variable with mean equal to 70
inches.
(a) What can be said about the probability that this year’s snowfall will be
between 55 and 85 inches?
(b) Can your answer be improved if, in addition, the standard deviation is known
to be 10 inches?

4.20 The number X of airplanes arriving at an airport during a given period of time is
distributed according to

Use the Chebyshev inequality to determine a lower bound for probability

during this period of time.

4.21 For each joint distribution given in Problem 3.13 (page 71), determine mX ,mY ,^2 X ,
2
Y ,and XY^ of random variables X^ and Y.
4.22 In the circu it shown in Figure 4.6, the resistance R is random and uniformly

+

V

r 0

R

i

Figure 4.6 Circuit diagram for Problem 4.22

Expectations and Moments 115

a) EfXjYˆygˆEfXgifXandYare independent.

b)EfXYjYˆygˆyEfXjYˆyg.

c) EfXYgˆEfYE[XjY]g.

x2.

PjX 1 j 0 :75)

PjXmXjh)

PjXmXjh)ash

P…Xa†

mX

a

:

pX…k†ˆ

100 ke^100

k!

; kˆ 0 ; 1 ; 2 ;...:

P80X120)



 

distributedbetween900 and 1100. The currentiˆ 0 :01A and the resistance

r 0 ˆ 1000

 are constants.