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Homework 3

Oct 3 (early issued Oct 1), 2024, due by 11:59pm Oct 10, 2024

1. (Expectations) Do problems 2.14, 2.17, 2.22, 2.26.

2. (Moments) Do problems 2.30, 2.38, 2.39.

3. Let X be a standard normal variable, i.e., X has pdf fX(x) = √2π/1 e −x 2/2 for all x ∈ (−∞, +∞). Define a discrete random variable Y by P(Y = √ 3) = P(Y = − √ 3) = 1/6 and P(Y = 0) = 2/3.

(i) Show that X and Y have the same r-th moment, for each r = 1, 2, 3, 4, 5.

(ii) Find another random variable Z which has the same r-th moments as X and Y for all r = 1, . . . , 5.