Question: (a) The gamma random variable X has probability density function given by f (x) =- where x > 0a>0b>0 bar(a) Derive EX(X - 1)] (4

(a) The gamma random variable X has probability density function given by f (x) =- where x > 0a>0b>0 bar(a) Derive EX(X - 1)] (4 Marks) (ii) the moment generating function, M(t) (5 Marks) Var(ax + b) (4 Marks) (b) The exponential random variable T has probability density function given by f(x) = = where t > 0 b > 0 What is the relationship between the gamma distribution and the exponential distribution? (1 Mark) (ii) Therefore give the expressions for E(T) and Var(T) (1 + 1 Marks) If the time in minutes for the service of a random customer to come completion follows has exponential distribution with expected time 10 minutes find the probability that for the service of a given customer is complete before 4 minutes. (4 Mark)

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