Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.5. The Multivariate Normal Distribution 209

3.5.12.LetXandY have a bivariate normal distribution with parametersμ 1 =
μ 2 =0,σ^21 =σ 22 = 1, and correlation coefficientρ. Find the distribution of the
random variableZ=aX+bYin whichaandbare nonzero constants.
3.5.13.Establish formula (3.5.11) by a direct multiplication.
3.5.14.LetX=(X 1 ,X 2 ,X 3 ) have a multivariate normal distribution with mean
vector 0 and variance-covariance matrix

Σ=

⎡

⎣

100

021

012

⎤

⎦.

FindP(X 1 >X 2 +X 3 +2).
Hint: Find the vectoraso thataX=X 1 −X 2 −X 3 and make use of Theorem
3.5.2.

3.5.15.SupposeXis distributedNn(μ,Σ). LetX=n−^1

∑n

i=1Xi.

(a)WriteXasaXfor an appropriate vectoraand apply Theorem 3.5.2 to find

the distribution ofX.

(b)Determine the distribution ofXif all of its component random variablesXi

have the same meanμ.

3.5.16. SupposeXis distributedN 2 (μ,Σ). Determine the distribution of the
random vector (X 1 +X 2 ,X 1 −X 2 ). Show thatX 1 +X 2 andX 1 −X 2 are independent
if Var(X 1 )=Var(X 2 ).

3.5.17.SupposeXis distributedN 3 ( 0 ,Σ), where

Σ=

⎡

⎣

321

221

113

⎤

⎦.

FindP((X 1 − 2 X 2 +X 3 )^2 > 15 .36).

3.5.18.LetX 1 ,X 2 ,X 3 be iid random variables each having a standard normal

distribution. Let the random variablesY 1 ,Y 2 ,Y 3 be defined by

X 1 =Y 1 cosY 2 sinY 3 ,X 2 =Y 1 sinY 2 sinY 3 ,X 3 =Y 1 cosY 3 ,

where 0≤Y 1 <∞, 0 ≤Y 2 < 2 π, 0 ≤Y 3 ≤π. Show thatY 1 ,Y 2 ,Y 3 are mutually
independent.
3.5.19.Show that expression (3.5.9) is true.
3.5.20.Prove Theorem 3.5.5.
3.5.21.SupposeXhas a multivariate normal distribution with mean 0 and covari-
ance matrix

Σ=

⎡

⎢

⎢

⎣

283 215 277 208

215 213 217 153

277 217 336 236

208 153 236 194

⎤

⎥

⎥

⎦.