STAT 714, FALL 2008 HOMEWORK 8
1. Let y= (y1, y2, y3)0∼ N3(µ,V), where
µ=
6
−2
1
and V=
1 0 −1
0 2 1
−1 1 3
.
Define the statistics x1= 3y1+y2−2y3and x2=y1−5y2+y3.
(a) Find the distribution of x= (x1, x2)0.
(b) Differentiate the mgf of xto find E(x) and cov(x). These should match your answers
in (a).
2. Suppose that φ(·) is the standard normal density function.
(a) For b < 1/2, compute
I(a, b) = ZR
eaz+bz2φ(z)dz.
(b) (↑) Suppose further that y1and y2are independent random variables with moment
generating functions mY1(t) = I(t, 0) and mY2(t) = I(0, t). Find the constant kso that
the statistic w=k(y2
1+y2) follows an exponential distribution with mean 2.
3. Let uand vhave the joint density
fU,V (u, v) = (π−1e−1
2(u2+v2), uv ≥0
0, uv < 0.
(a) Show that uand vare both marginally standard normal.
(b) Is the covariance matrix of (u, v)0singular or nonsingular?
(c) Does (u, v)0have a bivariate normal distribution? What is the main point?
4. Suppose that y∼ Np(µ,V), where r(V) = p, and let Abe a symmetric matrix
defining the quadratic form y0Ay. Let λ1, λ2, ..., λpdenote the eigenvalues of V1/2AV1/2.
(a) Give the conditions, in terms of λ1, λ2, ..., λp, under which y0Ay has a χ2distribution.
(b) When y0Ay does not have a χ2distribution, it is sometimes approximated by the
distribution of a constant multiple of a χ2random variable. The constant and the de-
grees of freedom are chosen to match the first two moments of y0Ay. Let udenote a
random variable having a (central) χ2distribution with rdegrees of freedom. Determine
the constants cand rin terms of λ1, λ2, ..., λpso that y0Ay and cu have the same mean
and variance.
5. Suppose that z1, z2, ..., znare iid standard normal random variables.
(a) Derive the joint distribution of z, z1−z, z2−z, ..., zn−z.
(b) (↑) Deduce that zand Pn
i=1(zi−z)2are independent.
(c) Let yhave an n-variate normal distribution with mean µand covariance matrix V,
where var(yi) = σ2, for all i, and cov(yi, yj) = σ2(1 −ρ), for i6=j, where 0 <ρ<1.
Prove that yand Pn
i=1(yi−y)2are independent. This is a generalization of part (b).
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