Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

212 Some Special Distributions

aspt(2.0,15), while the commandqt(.975,15)returns the 97.5th percentile of

this distribution. The R codet=seq(-4,4,.01)followed by plot(dt(t,3)~t)

yields a plot of thet-pdf with 3 degrees of freedom.

Before the age of modern computing, tables of the distribution ofTwere used.

Because the pdf ofT does depend on its degrees of freedomr, the usualt-table

gives selected quantiles versus degrees of freedom. Table III in Appendix D is such

a table. The following three lines of R code, however, produce this table.

ps = c(.9,.925,.950,.975,.99,.995,.999); df = 1:30; tab=c()

for(r in df){tab=rbind(tab,qt(ps,r))}; df=c(df,Inf);nq=qnorm(ps)

tab=rbind(tab,nq);tab=cbind(df,tab)

This code is the body of the R functionttablefound at the site listed in the

Preface. Due to the fact thatt-distribution converges to theN(0,1) distribution,

only the degrees of freedom from 1 to 30 are used in such tables. This is, also, the

reason that the last line in the table are the standard normal quantiles.

Remark 3.6.1.Thet-distribution was first discovered by W. S. Gosset when he

was working for an Irish brewery. Gosset published under the pseudonym Student.

Thus this distribution is often known asStudent’st-distribution.

Example 3.6.1(Mean and Variance of thet-Distribution).Let the random variable
Thave at-distribution withrdegrees of freedom. Then, as in (3.6.1), we can write
T=W(V/r)−^1 /^2 ,whereW has aN(0,1) distribution,Vhas aχ^2 (r) distribution,
andW andVare independent random variables. Independence ofW andVand
expression (3.3.8), provided (r/2)−(k/2)>0 (i.e.,k<r), implies the following:

E(Tk)=E

[

Wk

(

V

r

)−k/ 2 ]

=E(Wk)E

[(

V

r

)−k/ 2 ]

(3.6.3)

= E(Wk)

2 −k/^2 Γ

(r

2 −

k

2

)

Γ

(r

2

)

r−k/^2

ifk<r. (3.6.4)

BecauseE(W)=0,themeanofTis 0, as long as the degrees of freedom ofTexceed

1. For the variance, usek= 2 in expression (3.6.4). In this case the conditionr>k
   becomesr>2. SinceE(W^2 ) = 1, by expression (3.6.4), the variance ofTis given
   by

Var(T)=E(T^2 )=

r

r− 2

. (3.6.5)

Therefore, at-distribution withr>2 degrees of freedom has a mean of 0 and a

variance ofr/(r−2).

3.6.2 TheF-distribution........................

Next consider two independent chi-square random variablesUandVhavingr 1 and
r 2 degrees of freedom, respectively. The joint pdfh(u, v)ofUandV is then

h(u, v)=

{

1

Γ(r 1 /2)Γ(r 2 /2)2(r^1 +r^2 )/^2 u

r 1 / 2 − (^1) vr 2 / 2 − (^1) e−(u+v)/ (^20) <u,v<∞
0elsewhere.