Question

Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for i = 1, 2, 3 (that is, each is normally distributed with mean mean E(Y) = i and variance V(X) = i2). For each of the following situations, use the Y, i = 1, 2, 3 to construct a statistic with the indicated distribution a) X2 with 3 degrees of freedom b) t distribution with 2 degrees of freedom c) F distribution with 1 and 2 degrees of freedom

Answer 1

$Y_{i} \sim N( i, i^{2})$

E(Y₁) = 1 E(Y2) =2, E(Y3) =3

Var(Y₁) =1, Var (Y₂) =4, Var(Y₃) =9

$Z_{1} = \frac{Y_{1}-1}{1} \sim N(0,1)$

$Z_{2} = \frac{Y_{2}-2}{2} \sim N(0,1)$

$Z_{3} = \frac{Y_{3}-3}{3} \sim N(0,1)$

The square of standard normal variate is chi-square variate

$Z_{i}^{2} \sim \chi ^{2}_{1}$   i=1 :3

a) Chi- square with three degrees of freedom

U = $\sum_{i=1}^{3}Z_{i}^{2} \sim \chi ^{2}_{3}$

The sum of chi-square variate is again chi-square variate

b) t-distributon with 2 degrees of freedom

$V =\frac{Z_{1}}{\sqrt{\frac{Z_{2}^{2}+ Z_{3}^{2}}{2}}}$

$Z_{2}^{2} + Z_{3}^{2} \sim \chi ^{2}_{2}$

V is the ratio of standard normal variate to chi-square variate divided by its degrees of freedom which follows t distribution with 2 degrees of freedom.

$V =\frac{Z_{1}}{\sqrt{\frac{Z_{2}^{2}+ Z_{3}^{2}}{2}}} \sim t_{2}$

c) F -distribution with 1 and 2 degrees of freedom.

$W = \frac{Z_{1}^{2}}{\frac{Z_{2}^{2}+ Z_{3}^{2}}{2}}$

W is the ratio of two chi-square variate divided by its degrees of freedom which follows F distribution with 1 and 2 degrees of freedom.

$W = \frac{Z_{1}^{2}}{\frac{Z_{2}^{2}+ Z_{3}^{2}}{2}} \sim F _{1,2}$