Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.3. TheΓ,χ^2 ,andβDistributions 175

and

M′′(t)=(−α)(−α−1)(1−βt)−α−^2 (−β)^2.

Hence, for a gamma distribution, we have

μ=M′(0) =αβ

and

σ^2 =M′′(0)−μ^2 =α(α+1)β^2 −α^2 β^2 =αβ^2.

SupposeXhas a Γ(α, β) distribution. To calculate probabilities for this distri-

butioninR,leta=αandb=β. Then the commandpgamma(x,shape=a,scale=b)

returnsP(X≤x), while the value of the pdf ofXatxis returned by the command

dgamma(x,shape=a,scale=b).

Example 3.3.1.LetXbe the lifetime in hours of a certain battery used under

extremely cold conditions. SupposeXhas a Γ(5,4) distribution. Then the mean life-

time of the battery is 20 hours with standard deviation

√

5 ×16 = 8.94 hours. The

probability that battery lasts at least 50 hours is1-pgamma(50,shape=5,scale=4)

=0.0053. The median lifetime of the battery isqgamma(.5,shape=5,scale=4)

=18.68 hours. The probability that the lifetime is within one standard deviation

of its mean lifetime is

pgamma(20+8.94,shape=5,scale=4)-pgamma(20-8.94,shape=5,scale=4)=.700.

Finally, this line of R code presents a plot of the pdf:

x=seq(.1,50,.1); plot(dgamma(x,shape=5,scale=4)~x).

On this plot, the reader should locate the above probabilities and the mean and

median lifetimes of the battery.

The main reason for the appeal of the Γ-distribution in applications is the variety
of shapes of the distribution for different values ofαandβ. This is apparent in
Figure 3.3.1 which depicts six Γ-pdfs.^4
SupposeXdenotes the failure time of a device with pdff(x)andcdfF(x). In
practice, the pdf ofXis often unknown. If a large sample of failure times of these
devices is at hand then estimates of the pdf can be obtained as discussed in Chapter

4. Another function that helps in identifying the pdf ofXis thehazard function
   ofX.Letxbe in the support ofX. Suppose the device has not failed at timex,
   i.e.,X>x. What is the probability that the device fails in the next instance? We
   answer this question in terms of the rate of failure atx,whichis:

r(x) = lim

Δ→ 0

P(x≤X<x+Δ|X≥x)

Δ

=

1

1 −F(x)

lim

Δ→ 0

P(x≤X<x+Δ)

Δ

=

f(x)

1 −F(x)

. (3.3.3)

The rate of failure at timex,r(x), is defined as thehazard functionofXatx.

(^4) The R function for these plots is newfigc3s3.1.R, at the site listed in the Preface.