Question: Let N(t) be a Poisson process with rate and Sn be the time (starting from time 0) of the nth arrival, n = 1, 2,

Let N(t) be a Poisson process with rate and Sn be the time (starting from time 0) of the nth arrival, n = 1, 2, 3, .....

(i) Find the Probability [ S1 > s1, S2 > s2 ], where 0 < s1 < s2

(ii) Differentiate partially the expression you obtained as the answer in (i) with respect to s1 first and then with respect to s2. Simplify the resulting expression. This is the joint density function of S1 and S2

(iii) Using the expression you obtained in (ii), obtain the marginal probability density function of S1

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