By Olive D.J.

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For example, the weight of 5 feet tall students is less than the weight of 6 feet tall students, on average. Notation: When computing E(h(Y )), the marginal pdf or pmf f(y) is used. When computing E[h(Y )|X = x], the conditional pdf or pmf f(y|x) is used. In a formula such as E[E(Y |X)] the inner expectation uses f(y|x) but the outer expectation uses f(x) since E(Y |X) is a function of X. In the formula below, we could write EY (Y ) = EX [EY |X (Y |X)], but such notation is usually omitted. CHAPTER 2.

18. , n so that the moment generating function of X is n etxf(x). MX (t) = x=0 d MX (t) = MX (t). dt b) Find MX (0). d2 c) Find MX (t) = 2 MX (t). dt d) Find MX (0). dk e) Find M (k) (t) = k MX (t). 19. Suppose that the random variable W = eX where X ∼ N(µ, σ 2). Find E(W r ) = E[(eX )r ] by recognizing the relationship of E[(eX )r ] with the moment generating function of a normal(µ, σ 2) random variable. 20. Let X ∼ N(µ, σ 2 ) so that EX = µ and Var X = σ 2. a) Find E(X 2 ). b) If k ≥ 2 is an integer, then E(X k ) = (k − 1)σ 2E(X k−2 ) + µE(X k−1 ).

B) Find the moment generating function MX (t) of X. dn MX (t). (Hint: ﬁnd M (n) (t) for n = 1, 2, and 3. 28. 5. Find the moment generating function MGF of X. 29. Let W have a lognormal (µ, σ 2) distribution. Then W = eX where X ∼ N(µ, σ 2 ). Find E(W r ) = E[(eX )r ] by recognizing the relationship of E[(eX )r ] with the moment generating function of a normal (µ, σ 2 ) random variable. CHAPTER 1. 30. Use the kernel method to ﬁnd E(Y r ) for the chi (p, σ) distribution. , Yn that are of interest.