3. Let X be a random variable such that E(X^2k) = (2k)!/(2^k k!), E(X^2k+1) = 0, k = 0, 1, 2,. Find the moment generating function (m.g.f.) of X in terms of the moments given and the probability distribution function (p.d.f.) of X. (10%)

5.The lifetime of an automobile battery is described by an r.v. \( X \) having normal distribution with unknown mean \( \mu \) hour and known s.d. \( \sigma=150 \) hour. Let \( X_1, X_2, \cdots, X_{25} \) be independent r.v.'s distributed as \( X \) and suppose that \( \bar{x}=1725 \) hour. Test the hypothesis \( H_0: \mu=1800 \) versus \( H_1: \mu<1800 \) at level of significance \( \alpha=0.01 \) and make a conclusion. (10%)

3. Let \( X \) be a random variable such that \( E(X^{2k}) = \frac{(2k)!}{2^k k!} \), \( E(X^{2k+1}) = 0 \), \( k = 0, 1, 2, \dots \). Find the moment generating function (m.g.f.) of \( X \) in terms of the moments given and the probability distribution function (p.d.f.) of \( X \). (10%)

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