178 Some Special DistributionsThat is, the waiting time until thekthevent,Wk, has the gamma distribution with
α=kandβ=1/λ.LetT 1 be the waiting time until the first event occurs, i.e.,
k= 1. Then the pdf ofT 1 is
fT 1 (w)={
λe−λw 0 <w<∞
0elsewhere. (3.3.6)Hence,T 1 has the Γ(1, 1 /λ)-distribution. The mean ofT 1 =1/λ, while the mean
ofX 1 isλ. Thus, we expectλevents to occur in a unit of time and we expect the
first event to occur at time 1/λ.
Continuing in this way, fori≥1, letTidenote the interarrival time of theith
event; i.e.,Tiis the time between the occurrence of event (i−1) and eventi.As
shownT 1 has the Γ(1, 1 /λ). Note that Axioms (1) and (2) of the Poisson process
only depend onλand the length of the interval; in particular, they do not depend
on the endpoints of the interval. Further, occurrences in nonoverlapping intervals
are independent of one another. Hence, using the same reasoning as above,Tj,
j ≥2, also has the Γ(1, 1 /λ)-distribution. Furthermore,T 1 ,T 2 ,T 3 ,...are inde-
pendent. Note the waiting time until thekthevent satisfiesWk=T 1 +···+Tk.
Thus by Theorem 3.3.1,Wkhas a Γ(k, 1 /λ) distribution, confirming the derivation
above. Although this discussion has been intuitive, it can be made rigorous; see,
for example, Parzen (1962).
3.3.1 Theχ^2 -Distribution
Let us now consider a special case of the gamma distribution in whichα=r/2,
whereris a positive integer, andβ= 2. A random variableXof the continuous
type that has the pdff(x)={ 1
Γ(r/2)2r/^2 xr/ 2 − (^1) e−x/ (^20) <x<∞
0elsewhere,
(3.3.7)
and the mgf
M(t)=(1− 2 t)−r/^2 ,t<^12 ,
is said to have achi-square distribution(χ^2 -distribution), and anyf(x)ofthis
form is called achi-square pdf. The mean and the variance of a chi-square dis-
tribution areμ=αβ=(r/2)2 =randσ^2 =αβ^2 =(r/2)2^2 =2r, respectively. We
call the parameterrthe number of degrees of freedom of the chi-square distribution
(or of the chi-square pdf). Because the chi-square distribution has an important role
in statistics and occurs so frequently, we write, for brevity, thatXisχ^2 (r)tomean
that the random variableXhas a chi-square distribution withrdegrees of freedom.
The R functionpchisq(x,r)returnsP(X≤x) and the commanddchisq(x,r)
returns the value of the pdf ofXatxwhenXhas a chi-squared distribution with
rdegrees of freedom.
Example 3.3.2.SupposeXhas aχ^2 -distribution with 10 degrees of freedom. Then
the mean ofXis 10 and its standard deviation is
√
20 = 4.47. Using R, its median
and quartiles areqchisq(c(.25,.5,.75),10)=(6. 74 , 9. 34 , 12 .55). The following