Stat 5101 Lecture Notes - School of Statistics

2.5. PROBABILITY THEORY AS LINEAR ALGEBRA 552-22. Suppose X 1 , ..., X n are exchangeable random variables. Show that− 1n − 1 ≤ cor(X i,X j ).Hint: Consider var(X 1 + ···+X n ). Compare with the preceding problem.2-23. An infinite sequence of random variables X 1 , X 2 , ... is said to be exchangeableif the finite sequence X 1 , ..., X n is exchangeable for each n.(a)(b)Show that correlations cor(X i ,X j ) for an exchangeable infinite sequencemust be nonnegative. Hint: Consider Problem 2-22.Show that the following construction gives an exchangeable infinite sequenceX 1 , X 2 , ...of random variables having any correlation in the range0 ≤ ρ ≤ 1. Let Y 1 , Y 2 , ... be an i. i. d. sequence of random variables withvariance σ 2 , let Z be a random variable independent of all the Y i withvariance τ 2 , and define X i = Y i + Z.2-24. Consider an infinite sequence of random variables X 1 , X 2 , ... havingcovariancescov(X i ,X j )=ρ |i−j| σ 2where −1