Page 132 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 115

times equally likely to choose either direction, determine the average time interval
(in minutes) that the miner will be trapped.
4.15 Show that:
a) EfXjY ˆ ygˆ EfXg if X and Y are independent.
b) EfXYjY ˆ ygˆ yEfXjY ˆ yg.
c) EfXYgˆ EfYE[XjY]g.
4.16 Let random variable X be uniformly distributed over interval 0 x 2. Deter-
mine a lower bound for P jX 1j 0:75) 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 P jX m X j h) as a func-
tion of h and compare it with its lower bound as determined by the Chebyshev
inequality. Show that the lower bound becomes a better approximation of
P jX m X j h)as h becomes large.
4.18 Let a random variable X take only nonnegative values; show that, for any a > 0,

m X
P X a† :
a
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
k 100
100 e
p k†ˆ ; k ˆ 0; 1; 2; .. . :
X
k!
Use the Chebyshev inequality to determine a lower bound for probability
P 80 X 120) during this period of time.
4.21 For each joint distribution given in Problem 3.13 (page 71), determine m X , m Y , 2 X ,
2 , and XY of random variables X and Y .

Y
4.22 In the circuit shown in Figure 4.6, the resistance R is random and uniformly
distributed between 900 and 1100
. The current i ˆ 0:01 A and the resistance
r 0 ˆ 1000
are constants.

r 0
+
V i
– R

Figure 4.6 Circuit diagram for Problem 4.22

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