1. Let X_{1}, · · · ,X_{n} be n (≥ 2) independent random variables, where each X_{i} has the exponential distribution with parameter λ_{i} > 0. Let Y_{n} = X_{1} + · · · + X_{n}.

(a) Find the probability density function of Y_{2}.

(b) Assume now that all λ_{i} are the same, say λ_{i} = λ_{,} for i = 1, · · · , n. Show, by induction, that Y_{n} is a Gamma(n, λ) random variable.

2. A die has its six faces painted in six different colours. You throw it repeatedly. Let N be the minimum number of throws required for all six colours to have appeared at least once. For i = 1, · · · , 6, let Ni be the number of throws you make until the i^{th} colour appears after i-1 different colours have already appeared.

Clearly, N_{1} = 1 and N_{2},N_{3} · · · ,N_{6} are mutually independent random variables.

(a) For each i = 2, · · · , 6, determine the probability mass function of N_{i}.

(b) Find the mean and variance of N_{i} for i = 2, · · · , 6.

(c) By expressing N in terms of N_{1}, · · · ,N_{6}, show that

3. If X is a positive random quantity with probability density function f_{X}(x) = αβx^{β-1} exp(-αx^{β}), x ≥ 0, where α and β are two positive constants, show that Y = X^{β} has an exponential distribution and ?nd its parameter.

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