356 CHAPTER 6 Inferential Statistics

(b) Write a TI program to generate 200 samples ofY.

(c) Graph the histogram generated in (b) simultaneously with the

density curve forY.

7. LetZ=rand^2 as in Exercise 5. Show that the density function for
   Zis given by

f(x) =







1

2 √x if 0< x≤^1 ,

0 otherwise.

8. We have seen that the density function for the normally-distributed
   random variableX having mean 0 and standard deviation 1 is

f(x) =

√^1

2 π

e−x

(^2) / 2
.
The χ^2 random variablewith one degree of freedom is the
random variable X^2 (whence the notation!). Using the ideas de-
veloped above, show that the density function forX^2 is given by
g(x) =

1

√

2 π

x−^1 /^2 e−x/^2.

(More generally, theχ^2 distribution with n degrees of free-

dom is the distribution of the sum ofn independentχ^2 random

variables with one degree of freedom.)^15

Below are the graphs of χ^2 with one and with four degrees of

freedom.

(^15) The density function for theχ (^2) distribution withndegrees of freedom turns out to be
g(x) = x
n/ 2 − (^1) e−x/ 2
2 n/^2 Γ(n 2 ) ,
where Γ(n 2 ) =(n 2 − 1 )! isnis even. Ifn= 2k+ 1 is odd, then
Γ
(n
2
)
= Γ
Å
k+^12
ã

Å
k−^12
ãÅ
k−^32
ã
···^32 ·^12 ·√π.