Let (X(1) : t > 0) be a Poisson process with intensity A > 0 and...

 

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Let (X(1) : t > 0) be a Poisson process with intensity A > 0 and X(0) = 0. Let 0 = to < ti < t2 < < tm be a set of fixed time points. For i = 1,...,m, let X, = X((t-1,t]) be the increment on the i-th interval. (a) Find the joint distribution of (X1,.. Xm) conditioned on X(tm) = n for integer n >0. (b) Suppose customers arrive at a store according to a Poisson process with intensity A = 2 per hour. Given that there were 12 customers that arrived between 10am and 2pm, how many customers arrived on average between 9am and 1lam? Let (X(1) : t > 0) be a Poisson process with intensity A > 0 and X(0) = 0. Let 0 = to < ti < t2 < < tm be a set of fixed time points. For i = 1,...,m, let X, = X((t-1,t]) be the increment on the i-th interval. (a) Find the joint distribution of (X1,.. Xm) conditioned on X(tm) = n for integer n >0. (b) Suppose customers arrive at a store according to a Poisson process with intensity A = 2 per hour. Given that there were 12 customers that arrived between 10am and 2pm, how many customers arrived on average between 9am and 1lam?