1. Let X1, · · · ,Xn be n (≥ 2) independent random variables, where each Xi has the exponential distribution with parameter λi > 0. Let Yn = X1 + · · · + Xn.
(a) Find the probability density function of Y2.
(b) Assume now that all λi are the same, say λi = λ, for i = 1, · · · , n. Show, by induction, that Yn 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 ith colour appears after i-1 different colours have already appeared.
Clearly, N1 = 1 and N2,N3 · · · ,N6 are mutually independent random variables.
(a) For each i = 2, · · · , 6, determine the probability mass function of Ni.
(b) Find the mean and variance of Ni for i = 2, · · · , 6.
(c) By expressing N in terms of N1, · · · ,N6, show that
3. If X is a positive random quantity with probability density function fX(x) = αβxβ-1 exp(-αxβ), x ≥ 0, where α and β are two positive constants, show that Y = Xβ has an exponential distribution and ?nd its parameter.