47 Let X be a normal random variable with mean and variance Use the results of Theoretical Exercise 46 to show that E X n (n 2j) ( (n 2j)) ((2j)) 2j j 0 OCR End In the equation above, 1 2 is the largest integer less than or equal to n 2 Check your answer by letting 1 and n 2

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Question: 47. Let X be a normal random variable with mean and variance . Use the results of Theoretical Exercise 46 to show that E[X^n]

47. Let X be a normal random variable with mean µ and variance σ². Use the results of Theoretical Exercise 46 to show that E[X^n] = ∑ (n/2j) (µ^(n-2j)) (σ²(2j)) / 2j!

j=0

--- OCR End ---

In the equation above, [1/2] is the largest integer less than or equal to n/2.
Check your answer by letting 1 and n = 2.

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