Download MATH 408 SPRING 2010 1. Suppose X1,X2,...,Xn are independent

MATH 408 SPRING 2010
QUIZ 3
1. Suppose X1 , X2 , . . . , Xn are independent normal random variables with
Pn Xi −µ 2
mean µ and variance σ 2 . Let Wn =
. What is the disi=1
σ
tribution of Wn ?
Solution.
Xi − µ
= Z ∼ N (0, 1)
σ
so
n
X
Wn =
Zi2 ∼ χ2n
i=1
i.e. Wn is the sum of n standard normals squared, so it is chi-squared with
n degrees of freedom.
2
) distribution and Y1 , . . . , Ym
2. Let X1 , . . . , Xn be a sample from an N (µX , σX
be an independent random sample from a N (µY , σY2 ) distribution. Show
2
/SY2 > c) (You may leave your
how to use the F distribution to find P (SX
answer in the form of a probability involving a random variable with F
distribution, but be sure to include the degrees of freedom).
(HINT:
a. (n − 1)S 2 /σ 2 has a χ2n−1 distribution with n − 1 degrees of freedom.
b. The random variable W has F distribution with m and n degrees of
freedom if
U/m
W =
,
V /n
where U and V are independent chi-square random variables with m
and n degrees of freedom, respectively.)
From part a of the hint, we have
2
(n − 1)SX
∼ χ2n−1
2
σX
(m − 1)SY2
∼ χ2m−1 .
σY2
From part b of the hint we know that
2
(n−1)SX
/(n
2
σX
2
(m−1)SY
2
σY
− 1)
/(m − 1)
= W ∼ Fn−1,m−1 .
Now we simplify W to obtain
W =
2 2
SX
σY
2
2 ,
SY σX
1
2
QUIZ 3
hence
P
2 2
2
SX
SX σY
σY2
>c
= P
2 > c σ2
XY2
SY2 σX
X
σY2
= P W >c 2 ,
σX
where W ∼ Fn−1,m−1 .