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

Question

1) Suppose that X is deterministically equal to 1. What is the expected total number of flashes (of either color) during the interval [0,2] ?

Expected total number of flashes:

2) Suppose that X=∞ (i.e., the blue bulb never dies). What is the expected value of the time of the first flash (of either color)?

Expected value of the time of the first flash:

3) In the time interval [0,X] , there are exactly 5 flashes. What is the probability that exactly 2 of them were red?

Probability that exactly 2 of the 5 flashes were red:

4) Suppose that X 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] .

(Enter e for the constant e . You may use standard notation for this numerical entry even though there will be no parser below the answer box. Enter an exact answer or a numerical answer accurate to at least 3 decimal places.)

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

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

MAP estimate of X :

Answer 1

1) 4
2) 1/3
3) 1/3
4) ?????
5) ?????

Answer 2

1) 4
2) 1/3
3) 80/243

Answer 3

0.13925

Answer 4

Hint for 3: see 9.23.7 and use combination and conditional probabilities.