(Solved): Question 3 Suppose that \( Z_{1}, Z_{2}, \ldots, Z_{n} \) are statistically i ...

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Question 3 Suppose that \( Z_{1}, Z_{2}, \ldots, Z_{n} \) are statistically independent random variables. Define \( Y \) as the sum of squares of these random variables: \[ Y=\sum_{i=1}^{n} Z_{i}^{2} \quad(n \geq 2) \] (a) Express the moment generating function \( M_{Y}(t) \) of the random variable \( Y \) in terms of moment generating functions involving the random variables \( Z_{i}^{2}, i=1, \ldots, n \). (b) Determine \( M_{Y}(t) \) for the special case that \( Z_{i} \sim N(0,1) \). (c) For the above special case, calculate \( E[Y] \) by using the moment generating function.