1 n 5. Let U, U2,..., U be independent and identically distributed random variables from Uniform...

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1 n 5. Let U, U2,..., U be independent and identically distributed random variables from Uniform (0,1). (a) Derive the probability density function of X = -log U. Which family of distributions does it belong to? [5 marks] (b) Use the result in part (a), or otherwise, to show that the moment generating function of Y = -log(UU ...U) is 1 My (t) = (1-1)" t <1, " and hence determine the distribution of Y. [5 marks] (c) Determine the probability density function of Z = UU U 2 [5 marks] (d) Express E(Z") in terms of n and r. [5 marks] 1 n 5. Let U, U2,..., U be independent and identically distributed random variables from Uniform (0,1). (a) Derive the probability density function of X = -log U. Which family of distributions does it belong to? [5 marks] (b) Use the result in part (a), or otherwise, to show that the moment generating function of Y = -log(UU ...U) is 1 My (t) = (1-1)" t <1, " and hence determine the distribution of Y. [5 marks] (c) Determine the probability density function of Z = UU U 2 [5 marks] (d) Express E(Z") in terms of n and r. [5 marks]