Consider a server with Poisson job-arrival stream at an average rate of $60$ per hour. Determine the probability that the time interval between successive job arrivals is
a. longer than $4$ min
b. shorter than $8$ min
c. between $2$ and $6$ min
What I've done so far:
I got $A= 0.01831$ and $B=0.9996$
I tried to do part c and get $P(X < 6)-P(X > 2)$ to get the interval $P$ but the answer I got was wrong. Can you help me understand what I did wrong?
Best Answer
Interarrival times $X$ are exponential with rate 60 per hour, or better for these problems $\lambda = 1$ per minute.
(a) $P\{X > 4\} = \exp (-4) = 0.01831,$ just as you said.
(b) $P\{X \leq 8\} = 1 - \exp(-8) = 0.9997,$ very close to your answer.
(c) $$P\{2 \le X \le 6\} = P\{X \le 6\} - P\{X \le 2\} = (1 - e^{-6}) - (1 - e^{-2}) = e^{-2} - e^{-6} = 0.1328565.$$ You had a backwards inequality in your attempt, otherwise OK.
Addendum (31 Mar '15): This illustrates a general principle. If $F$ is the CDF of $X$, then $P\{a < X \le b\} = F(b) - F(a).$