Stat 5101 Lecture Notes - School of Statistics

5.4. THE MULTINOMIAL DISTRIBUTION 1615-2. Is ⎛⎞⎝ 3 2 2 3 −1/32 ⎠−1/3 2 3a covariance matrix? If not, why not? If it is a covariance matrix, is a randomvector having this covariance matrix degenerate or non-degenerate?5-3. Consider the degenerate random vector (X, Y )inR 2 defined byX = sin(U)Y = cos(U)where U ∼U(0, 2π). We say that (X, Y ) has the uniform distribution on theunit circle. Find the mean vector and covariance matrix of (X, Y ).5-4. Let M be any symmetric positive semi-definite matrix, and denote itselements m ij . Show that for any i and j−1 ≤m ij√mii m jj≤ 1 (5.41)Hint: Consider w ′ Mw for vectors w having all elements zero except the i-thand j-th.The point of the problem (this isn’t part of the problem, just the explanationof why it is interesting) is that if M is a variance, then the fraction in (5.41) iscor(X i ,X j ). Thus positive semi-definiteness is a stronger requirement than thecorrelation inequality, as claimed in Section 5.1.4.5-5. Show that the usual formula for the univariate normal distribution is theone-dimensional case of the formula for the multivariate normal distribution.5-6. Show that a constant random vector (a random vector having a distributionconcentrated at one point) is a (degenerate) special case of the multivariatenormal distribution.5-7. Suppose X =(X 1 ,...,X k ) has the multinomial distribution with samplesize n and parameter vector p =(p 1 ,...,p k ), show that for i ≠ jvar(X i − X j )n= p i + p j − (p i − p j ) 25-8. If X ∼N(0, M) is a non-degenerate normal random vector, what is thedistribution of Y = M −1 X?5-9. Prove (5.35).Hint: WriteX 1 − µ 1 =[X 1 −E(X 1 |X 2 )] + [E(X 1 | X 2 ) − µ 1 ]then use the alternate variance and covariance expressions in Theorem 5.2 andlinearity of expectation.