Fundamentals of Probability and Statistics for Engineers

random variable representing distance of the hit from the center. Suppose that the

pdf of R is

Compute the mean score of each shot.

4.6 A random variable X has the exponential distribution

Determine:

(a) The value of a.

(b) The mean and variance of X.

(c) The mean and variance of Y (X/2) 1.

4.7 Let the mean and variance of X be m and^2 , respectively. For what values of a and b
does random variable Y, equal to aX b, have mean 0 and variance 1?

4.8 Suppose that your waiting time (in minutes) for a bus in the morning is uniformly
distributed over (0, 5), whereas your waiting time in the evening is distributed as
shown in Figure 4.4. These waiting times are assumed to be independent for any
given day and from day to day.
(a) If you take the bus each morning and evening for five days, what is the mean of
your total waiting time?
(b) What is the variance of your total five-day waiting time?
(c) What are the mean and variance of the difference between morning and evening
waiting times on a given day?
(d) What are the mean and variance of the difference between total morning wait-
ing time and total evening waiting time for five days?

fT(t)

010

t

Figure 4.4 Density function of evening waiting times, for Problem 4.8

Expectations and Moments 113

fR…r†ˆ

2

… 1 ‡r^2 †

; forr> 0 ;

0 ; elsewhere:

8

<

:

fX…x†ˆ ae

x= (^2) ; forx 0 ;
0 ; elsewhere:

ˆ 

‡