Page 237 - Probability and Statistical Inference

214 4. Functions of Random Variables and Sampling Distribution

} = + (n – 1)σ . This leads to the desired conclusion: }
2
= + (1 – 1/n)σ .
2
For completeness, we now give the pdf of the p-dimensional normal ran-
dom vector. To be specific, let us denote a p-dimensional column vector X
whose transpose is X’ = (X , ..., X ), consisting of the real valued random
1
p
th
variable X as its i component, i = 1, ..., p. We denote E[X ] = µ , V[X ] = σ ,
i
i
ii
i
i
and Cov(X , X ) = σ , 1 ≤ i ≠ j ≤ p. Suppose that we denote the mean vector
i
j
ij
µ′
th
µ µ µ µ µ where µ′ µ′µ′ µ′ = (µ , ..., µ ) and then write down σ as the (i, j) element of the
p
ij
1
p × p matrix ΣΣ ΣΣ Σ, 1 ≤ i ≠ j = p. Then, ΣΣ ΣΣ Σ = (σ ) is referred to as the variance-
ij p×p
covariance matrix or the dispersion matrix of the random vector X.
Assume that the matrix ΣΣ ΣΣ Σ has the full rank which is equivalent to saying
–1
that ΣΣ ΣΣ Σ exists. Then, the random vector X has the p-dimensional normal
distribution with mean vector µµ µµ µ and dispersion matrix ΣΣ ΣΣ Σ, denoted by N (µµ µµ µ,
p
Σ Σ Σ Σ Σ), provided that the joint pdf of X , ..., X is given by
p
1
where x = (x , ..., x ). One should check that the bivariate normal pdf from
1
p
(3.6.1) can be written in this form too. We leave it as the Exercise 4.6.3.
C. F. Gauss originally derived the density function given in (4.6.1) from
that of linear functions of independent normal variables around 1823-1826. In
many areas, including the sciences and engineering, the normal distributions
are also frequently referred to as the Gaussian distributions.

We leave the proofs of (4.6.2)-(4.6.3) as the Exercise 4.6.4.

Example 4.6.5 (Exercise 3.5.17 Continued) Suppose that the random
vector X = (X , X , X , X ) has the 4-dimensional normal distribution
2
1
4
3