Question

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 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]:

Answer 1

for time interval [0,2]

the red bulb flashes with rate 1*2 = 2

if X = 1 then blue light flash only in time interva; [0,1] thus the rate will be 2

if X = 2 then blue light can flash any time in interval [0,2] thus the rate will be 4

when x=1

p(exactly 3 arrival) = p(3 red flash)*p(0 blue flash) +p(2 red flash)* p(1 blue flash) +  p(1 red flash)*p(2 blue flash)

+p(3 blue flash)*p(0 red flash)

=0.18*0.135 + 0.073*0.271 + 0.271*0.271 + 0.135*0.18

=0.195

similarly when x=2

p(exactly 3 arrival) = 0.089

p(exactly 3 arrival) = p(x=1)*(pexactly 3/x=1) + p(x=2)*(pexactly 3/x=2)

=0.5*0.195+0.5*0.089

=0.142