Page 259 - Fundamentals of Probability and Statistics for Engineers

242 Fundamentals of Probability and Statistics for Engineers

7.24 Show that, if is a positive integer, the probability distribution function (PDF) of
a gamma-distributed random variable X can be written as
0; for x 0;
8
< 1 k x
F X x†ˆ P x† e
1 ; for x > 0:
: k!
kˆ0
Recognize that the terms in the sum take the form of the Poisson mass function and
therefore can be calculated with the aid of probability tables for Poisson distribu-
tions.
7.25 The system shown in Figure 7.16 has three redundant components, A–C. Let their
operating lives (in hours) be denoted by T 1 , T 2 , and T 3 , respectively. If the
redundant parts come into operation only when the online component fails (cold
redundancy), then the operating life of the system, T, is T ˆ T 1 ‡ T 2 ‡ T 3 .
Let T 1 , T 2 , and T 3 be independent random variables, each distributed as

1 e t j =100 ; for t j 0; j ˆ 1; 2; 3;
f t j †ˆ 100
T j
0; otherwise:
Determine the probability that the system will operate at least 300 hours.
7.26 We showed in Section 7.4.1 that an exponential failure law leads to a constant
failure rate. Show that the converse is also true; that is, if h(t) as defined by
Equation (7.65) is a constant then the time to failure T is exponentially distributed.
7.27 A shifted exponential distribution is defined as an exponential distribution shifted
to the right by an amount a; that is, if random variable X has an exponential
distribution with
x
e ; for x 0;
f x†ˆ
X
0; elsewhere;
random variable Y has a shifted exponential distribution if f (y) has the same
Y
shape as f (x) but its nonzero portion starts at point a rather than zero. Determine
X
the relationship between X and Y and probability density function (pdf) f (y).
Y
What are the mean and variance of Y ?
A

B

C

Figure 7.16 System of components, for Problem 7.25

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