Page 238 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 215

namely, N (µµ µµ µ, ΣΣ ΣΣ Σ) with µµ µµ µ = 0 and
4

with the matrices and vectors P 3×3 = 4I, Q’ 1×3 = (111) and S 1×1 = 1. The
matrix ΣΣ ΣΣ Σ is p.d. and it is easy to check that det(ΣΣ ΣΣ Σ) = 16.
Let us denote the matrices

Thus, applying (4.8.10), we have

and hence

In other words, in this particular situation, the pdf from (4.6.1) will reduce to

4
for (x , x , x , x ) ∈ ℜ . Thus, in the Exercise 3.5.17, the given random vector
2
3
4
1
X actually had this particular 4-dimensional normal distribution even though
at that time we did not explicitly say so.
Sampling Distributions: The Bivariate Normal Case
For the moment, let us focus on the bivariate normal distribution. Suppose
that (X , Y ), ..., (X , Y ) are iid where –∞ < µ , µ < ∞,
n
2
1
1
1
n
0 < < ∞ and –1 < ρ < 1, n ≥ 2. Let us denote