Stat 5101 Lecture Notes - School of Statistics

2.7. INDEPENDENCE 79Then X and Y are uncorrelated (Problem 2-37) but not independent. Independencewould require thatE{g(X)h(Y )} = E{g(X)}E{h(Y )}hold for all functions g and h. But it obviously does not hold when, to pick justone example, g is the squaring function and h is the identity function so g(X) =Y and h(Y )=Y, because no nonconstant random variable is independent ofitself. 2Problems2-27. Suppose X ∼ Bin(n, p).(a)Show thatE{X(X − 1)} = n(n − 1)p 2Hint: Follow the pattern of Example 2.5.1.(b)Show thatHint: Use part (a).var(X) =np(1 − p).2-28. Suppose X ∼ Poi(µ).(a)Show thatE{X(X − 1)} = µ 2Hint: Follow the pattern of Example 2.5.2.(b)Show thatHint: Use part (a).var(X) =µ.2-29. Verify the moments of the DU(1,n) distribution given in Section B.1.1of Appendix B.Hint: First establishn∑k 2 n(n + 1)(2n +1)=6by mathematical induction.k=12 Bizarrely, constant random variables are independent of all random variables, includingthemselves. This is just the homogeneity axiom and the “expectation of a constant is theconstant” property:E{g(a)h(X)} = g(a)E{h(X)} = E{g(a)}E{h(X)}for any constant a and random variable X.