Stat 5101 Lecture Notes - School of Statistics

5.4. THE MULTINOMIAL DISTRIBUTION 1635-13. For the random vector X defined by (5.23) in Example 5.1.3 suppose U,V , and W are i. i. d. standard normal random variables.(a)(b)What is the joint distribution of the two-dimensional random vector whosecomponents are the first two components of X?What is the conditional distribution of the first component of X given thesecond?5-14. Suppose Z 1 , Z 2 , ... are i. i. d. N (0,τ 2 ) random variables and X 1 , X 2 ,... are defined recursively as follows.• X 1 is a N (0,σ 2 ) random variable that is independent of all the Z i .• for i>1X i+1 = ρX i + Z i .There are three unknown parameters, ρ, σ 2 , and τ 2 , in this model. Becausethey are variances, we must have σ 2 > 0 and τ 2 > 0. The model is called anautoregressive time series of order one or AR(1) for short. The model is said tobe stationary if X i has the same marginal distribution for all i.(a)Show that the joint distribution of X 1 , X 2 , ..., X n is multivariate normal.(b) Show that E(X i ) = 0 for all i.(c)Show that the model is stationary only if ρ 2 < 1 andσ 2 = τ 21 − ρ 2Hint: Consider var(X i ).(d)Show thatcov(X i ,X i+k )=ρ k σ 2 , k ≥ 0in the stationary model.