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Included all steps. Thanks The random variable X is uniformly distributed in the interval [0, α]...

Included all steps. Thanks

The random variable X is uniformly distributed in the interval [0, α] for some α > 0.

Parameter α is fixed but unknown. In order to estimate α, a random sample X1, X2, . . . , Xn of independent and identically distributed random variables with the same distribution as X is collected, and the maximum value Y = max{X1, X2, ..., Xn} is considered as an estimator of α.

(a) Derive the cumulative distribution function of Y .

(b) Find the mean and variance of Y , and explain why Y is a good estimator for α when sample size n is large.

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