[SOLVED] Stat580 homework 2

1. Let X ∼ Np(0, I). Let Y be a random variable having the χ
2
p
-density that is independent of X. Let Z =
√
Y
X
kXk
. Show that that the density of Z is also standard
multivariate normal. [10 points]2. Write an example in C to illustrate that a function passes its arguments by value. To
do this, write a function which takes in two arguments: an integer and a pointer to
an integer and then increments each by 1. Follow the location of the arguments inside
and outside the function to illustrate the point. [10 points]3. Write a function in C which illustrates the use of the matrix multiplication algorithm
using CCS representation that you wrote in the previous assignment. [20 points]4. Let X1, X2, . . . , Xn be a random multivariate sample such that Xi has only the first
1 ≤ pi ≤ p coordinates that are observed. Assume that each Xi
is a realization from
the pi-dimensional marginal distribution of Np(µ, Σ).Further, there are at least two
is for which pi = p. Answer the following questions:
(a) Find the maximum likelihood estimator of µ and Σ using direct maximization of
the loglikelihood. [15 points](b) Use the expectation-maximization algorithm to formulate the maximum likelihood estimator of µ and Σ. [20 points](c) Compare the number of computations (floating point operations) needed in one
EM step to the number of computations in the direct calculations. You may make
simplifying assumptions as needed for calculating the number of operations. [20
points](d) How do the results in (c) change if the pi observed coordinates are not the first
ones? [5 points]