154 Multivariate Distributions2.8.9.Letμandσ^2 denote the mean and variance of the random variableX.Let
Y=c+bX,wherebandcare real constants. Show that the mean and variance of
Y are, respectively,c+bμandb^2 σ^2.
2.8.10.Determine the correlation coefficient of the random variablesXandY if
var(X)=4,var(Y) = 2, and var(X+2Y) = 15.2.8.11.LetXandYbe random variables with meansμ 1 ,μ 2 ; variancesσ 12 ,σ^22 ;and
correlation coefficientρ. Show that the correlation coefficient ofW=aX+b, a >0,
andZ=cY+d, c >0, isρ.
2.8.12. A person rolls a die, tosses a coin, and draws a card from an ordinary
deck. He receives $3 for each point up on the die, $10 for a head and $0 for a
tail, and $1 for each spot on the card (jack = 11, queen = 12, king = 13). If we
assume that the three random variables involved are independent and uniformly
distributed, compute the mean and variance of the amount to be received.
2.8.13.LetX 1 andX 2 be independent random variables with nonzero variances.
Find the correlation coefficient ofY=X 1 X 2 andX 1 intermsofthemeansand
variances ofX 1 andX 2.
2.8.14.LetX 1 andX 2 have a joint distribution with parametersμ 1 ,μ 2 ,σ^21 ,σ^22 ,
andρ. Find the correlation coefficient of the linear functions ofY=a 1 X 1 +a 2 X 2
and Z= b 1 X 1 +b 2 X 2 in terms of the real constantsa 1 ,a 2 ,b 1 ,b 2 ,andthe
parameters of the distribution.
2.8.15. LetX 1 ,X 2 ,andX 3 be random variables with equal variances but with
correlation coefficientsρ 12 =0. 3 ,ρ 13 =0.5, andρ 23 =0.2. Find the correlation
coefficient of the linear functionsY=X 1 +X 2 andZ=X 2 +X 3.
2.8.16.Find the variance of the sum of 10 random variables if each has variance 5
and if each pair has correlation coefficient 0.5.
2.8.17.LetXandYhave the parametersμ 1 ,μ 2 ,σ 12 ,σ^22 ,andρ. Show that the
correlation coefficient ofXand [Y−ρ(σ 2 /σ 1 )X] is zero.
2.8.18.LetS^2 be the sample variance of a random sample from a distribution with
varianceσ^2 >0. SinceE(S^2 )=σ^2 ,whyisn’tE(S)=σ?
Hint:Use Jensen’s inequality to show thatE(S)<σ.