Page 269 - Elements of Distribution Theory

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052184472Xc08 CUNY148/Severini May 24, 2005 17:54

8.6 Exercises 255

8.17 Let X 1 , X 2 ,..., X n denote independent random variables such that X j has a normal distribution
2
with mean 0 and variance σ > 0, j = 1,..., n, and let
j
1 n
n
2
T = (X j − ¯ X) , ¯ X = X j .
n
j=1 j=1
2
2
Find conditions on σ ,...,σ so that there exists a constant c such that cT has a chi-squared
1
n
distribution with r degrees of freedom. Give expressions for c and r.
8.18 Let Z 1 ,..., Z n denote independent random variables, each with a standard normal distribution,
and let δ 1 ,...,δ n denote real-valued constants.
Define a random variable X by
2
X = n
(Z j + δ j ) .
j=1
The distribution of X is called a noncentral chi-squared distribution with n degrees of freedom.
(a) Show that the distribution of X depends on δ 1 ,...,δ n only through
2 n
2
δ ≡ δ ;
j
j=1
2
δ is called the noncentrality parameter of the distribution.
(b) Find the mean and variance of X.
8.19 Suppose that X 1 and X 2 are independent random variables such that, for j = 1, 2, X j has a
noncentral chi-squared distribution with n j degrees of freedom, n j = 1, 2,..., and noncentrality
2
parameter γ ≥ 0. Does X 1 + X 2 have a noncentral chi-squared distribution? If so, find the
j
degrees of freedom and the noncentrality parameter of the distribution.
8.20 Let X denote a d-dimensional random vector with a multivariate normal distribution with mean
vector µ and covariance matrix , which is assumed to be positive-definite. Let A denote a
d × d symmetric matrix and consider the random variable
T
Q = X AX.
Find conditions on A so that Q has a noncentral chi-squared distribution. Find the degrees of
freedom and the noncentrality parameter of the distribution.
8.21 Let X = (X 1 ,..., X n ) denote a random vector such that X 1 ,..., X n are real-valued, exchange-
able random variables and suppose that X 1 has a standard normal distribution. Let

2 n
2
S = (X j − ¯ X) .
j=1
2
Let ρ = Cov(X i , X j ). Find the values of ρ for which S has a chi-squared distribution.
8.22 Let X = (X 1 ,..., X n ) denote a random vector such that X 1 ,..., X n are real-valued, exchange-
able random variables and suppose that X 1 has a standard normal distribution. Let
1 n
1 n
2
2
¯ X = X j and S = (X j − ¯ X) .
n n − 1
j=1 j=1
2
Find the values of ρ for which ¯ X and S are independent.
8.23 Let X denote a d-dimensional random vector with a multivariate normal distribution with mean
d
vector 0 and covariance matrix given by the identity matrix. For a given linear subspace of R ,
M, define
2
D(M) = min ||X − m||
m∈M