Fundamentals of Probability and Statistics for Engineers

7.24 Show that, if is a positive integer, the probability distribution function (PD F ) of
a gamma-distributed random variable X can be written as

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 ,andT 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 T 1 T 2 T 3.
Let T 1 ,T 2 ,andT 3 be independent random variables, each distributed as

Determine the probability that the system will operate at least 300hours.

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

random variable Y has a shifted exponential distribution if fY (y) has the same

shape as fX (x) but its nonzero portion starts at point a rather than zero. Determine

the relationship between X and Y and probability density function (pdf) fY (y).

What are the mean and variance of Y?

A

B

C

Figure 7.16 System of components, for Problem 7.25

242 Fundamentals of Probability and Statistics for Engineers



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