Page 261 - Probability and Statistical Inference

238 4. Functions of Random Variables and Sampling Distribution

by appealing to the Definition 4.6.1 of a multivariate normal distribution, show
that
is distributed as with some ρ .
*
{Hint: Look at (4.6.6).}
4.6.9 Suppose that (X , Y ), ..., (X , Y ) are iid with
n
n
1
1
and –1 < ρ < 1, n ≥ 3. Consider the
Pearson correlation coefficient r defined by (4.6.7). The pdf of r when ρ = 0
is given by

where Use transformation tech-
niques to derive the following result:

4.6.10 Suppose that X , ..., X are iid N (µ, ΣΣ ΣΣ Σ) where µ ∈ ℜ and ΣΣ ΣΣ Σ is a 2
2
2
1
n
× 2 p.d. matrix, n ≥ 2. Let us denote and
Prove the following sampling distributions:
(i) is distributed as N (µ, n ΣΣ ΣΣ Σ);
–1
2
–1
(ii) n( – µ)′ ΣΣ ΣΣ Σ ( – µ) is distributed as .
{Hint: Apply the Definition 4.6.1 in part (i). To prove part (ii), without any
loss of generality, assume that n = 1 and µµ µµ µ = 0. Now, show that n( – µ)′ ΣΣ ΣΣ Σ –
1 ( – µ) can be written as which is further split

up as . At this stage, can the re-
productive property of Chi-squares be applied?}
4.6.11 Suppose that the random vector X, where X′ = (X , ..., X ), has a p-
p
1
dimensional normal distribution with the mean vector 0 and the p × p p.d. disper-
sion matrix ΣΣ ΣΣ Σ, denoted by N (0, ΣΣ ΣΣ Σ). We assume that each diagonal entry in ΣΣ ΣΣ Σ is 1.
p
th
In other words, the (i, j) entry in the matrix ΣΣ ΣΣ Σ corresponds to the population
correlation coefficient ρ between the variables X , X , 1≤ i ≠ j = p. Suppose
i
j
ij
2
that Y is a positive real valued random variable distributed as χ . It is also
ν
assumed that Y and (X , ..., X ) are independent. Let us denote p new random
1
p
1/2
variables T = X ÷ (Y/ν) , i = 1, ..., p. Jointly, however, (T , ..., T ) is said to
p
i
i
1
have the p-dimensional t distribution which depends on the correlation matrix ΣΣ ΣΣ Σ,
–1
denoted by Mt (ν, ΣΣ ΣΣ Σ). Assume that ΣΣ ΣΣ Σ exists. Then, show that the joint
p