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4. Functions of Random Variables and Sampling Distribution 237

4.5.9 (Exercise 4.5.8 Continued) Let Z , Z be iid standard normal. Evalu-
2
1
ate where 0 < c < 1 is a fixed but arbitrary number.
4.5.10 Verify the expression of the variance of W, which is distributed as
t , given in (4.5.4).
í
4.5.11 Verify the expression of the mean and variance of U, which is
distributed as F í1,í2 , given in (4.5.11)-(4.5.12).
4.6.1 (Example 4.6.1 Continued) Use transformations directly to show
that (X , ) is distributed as when X , ..., X are iid
n
1
1
N(µ, σ ).
2
4.6.2 (Example 4.6.2 Continued) Use transformations directly to find the
bivariate normal distribution of (aX + bX , ) when X , ..., X are iid N(µ
1
2
1
n
σ ). Here, a and b are fixed non-zero real numbers.
2
4.6.3 Show that the bivariate normal density from (3.6.1) can be expressed
in the form given by (4.6.1).
4.6.4 Verify the properties given in (4.6.2)-(4.6.3) for the multivariate nor-
mal distribution.
4.6.5 Let X , ..., X be iid N(µ, σ ). Then, find the joint distributions of
2
n
1
(i) Y , ..., Y where Y = X – X , i = 2, ..., n;
2 n i i 1
(ii) U , ..., U where U = X – X , i = 2, ..., n.
2
i–1
n
i
i
{Hint: Use the Definition 4.6.1 for the multivariate normality.}
4.6.6 (Exercise 4.6.5 Continued) Solve the Exercise 4.6.5 by using direct
transformation techniques.
2
4.6.7 Suppose that X , ..., X are iid N(µ, σ ), n ≥ 2. Use the variables U ,
1 n 1
..., U where
n
(i) Show that U = (U , ..., U ) has the n-dimensional normal distri
1 n
bution;
(ii) Show that U and (U , ..., U ) are independent;
1 2 n
(iii) Use parts (i)-(ii) to derive the distribution of ;
2
(iv) Express the sample variance S as a function of U , ..., U alone.
n
2
Hence, show that and S are independently distributed.
2
{Hint: Use the Definition 4.6.1 for the multivariate normality to solve part (i).
Also, observe that can be rewritten as
4.6.8 Suppose that (X , Y ), ..., (X , Y ) are iid
1
1
n
n
with –∞ < µ , µ < ∞, 0 < < ∞ and –1 < ρ < 1, n ≥ 2. Then,
1
2