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Starting at time 0, a red bulb flashes according to a Poisson process with rate ?=1....

Starting at time 0, a red bulb flashes according to a Poisson process with rate ?=1. Similarly, starting at time 0, a blue bulb flashes according to a Poisson process with rate ?=2, but only until a nonnegative random time ?, at which point the blue bulb “dies." We assume that the two Poisson processes and the random variable ? are (mutually) independent.

Suppose that ? is equal to either 1 or 2, with equal probability. Write down an expression for the probability that there were exactly 3 arrivals during the time interval [0,2].

Probability that there were exactly 3 arrivals during the time interval [0,2]:

Suppose that ? is an exponential random variable with parameter (and mean) equal to 1. Find the MAP estimate of ?, given that there were exactly 5 blue flashes.

MAP estimate of ?

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