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The random variable Y has an exponential distribution with probability density function (pdf) as follows: f(y) = λe−λy,...

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Statistics

The random variable Y has an exponential distribution withprobability density function (pdf)

as follows:

f(y) = λe−λy, y >0

= 0, otherwise

(i) Showing your workings, find P (Y > s|Y > t), for s ≥t. [3]

(ii) Derive an expression for the conditional pdf of Y ,conditional on that Y ≤ 200. [3]

N(t) is a Poisson process with rate λ

(iii) Find an expression for the Cumulative DistributionFunction (CDF) of the waiting time until the first event. (Hint:Consider the probability of there being 0 events in time t.)[2]

(iv) Explain the relationship between the mean of the Poissondistribution with rate λ and the mean of the associateddistribution for the waiting time.

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