Let X1,……,X10 be independent N(-2,4) random variables, and let W be another random variable
independent of X1,…….,X10 Denote
Y: ∑10
𝑗=1 𝑋𝑗
1
∑10
(𝑋𝑗 + 2)^2
4 𝑗=1
And assume that W+U has a gamma distribution with parameters α=20/3 and λ=1/2
1) Find the distribution of Y and specify all numerical values of its parameters?
2) Find the distribution of U and specify all numerical values of its parameters?
3) Use the moment generating function to find the distribution of W. Specify all numerical
values of its parameters
Assume that X1,X2,….are random variables with a common finite mean μ and a common
finite variance σ^2. Assume that there exists r∈N, such that for any n≥ 1, Xn and
Xn+1,…..,Xn+r may be correlated, but Cov(Xn,Xn+k)=0 whenver k≥r+1. For any n∈N let X bar
be the sample mean of X1,…., Xn. Show that Xn bar m. s → μ and hence Xn bar p → μ.
3. Let X and Y be jointly normal random variables with μx=1 μy=-2 σ^2x=4 σ^2y=6 σxy=-3
1) Find the joint characteristics function of (X,Y)
2) Let V=2X=Y. Find the joint characteristic function of (V,Y)
3) Are V and Y jointly normal? Explain why or why not?
4. Let (Xj)j ∈ N be a sequence of independent and identically distributed uniform random
variables on [-a,a], for some a>0. Denote Yn:= ∑𝑛𝑗=1[Xj − E(Xj)]/√∑𝑛𝑗=1 𝑉𝑎𝑟(𝑋𝑗), n∈ N
Find the characteristic function of Yn for each n∈ N. Show that Yn converge in distribution,
as n→infinity, to a limiting random variable Y and identify the distribution of Y.