Phone calls are received at Janice’s house according to a Poisson Process with parameter λ=2 per hour.

How long can Janice’s shower be if she wishes the probability of receiving no phone calls to be at most 0.5?

What I have tried: Substituted λt in the poisson formula with x = 0 which should be less than 0.5. Should I be taking log on both sides?

The answer is 20 minutes - how?

  • $\begingroup$ Welcome to CV. Since you’re new here, you may want to take our tour, which has information for new users. Please add the [self-study] tag and read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. $\endgroup$ – jbowman Sep 9 '18 at 18:03
  • $\begingroup$ @jbowman Not my homework, it is entirely for self-study. Will try and show what I have done till now. $\endgroup$ – user218970 Sep 9 '18 at 18:04
  • $\begingroup$ My apologies, we get a lot of "please do my homework for me" questions. I'll modify my standard text to take into account the possibility that it really is self-study. $\endgroup$ – jbowman Sep 9 '18 at 18:06
  • $\begingroup$ @jbowman Thank you so much for being considerate. I'm sorry for not adding my attempt at solving this question into the post. $\endgroup$ – user218970 Sep 9 '18 at 18:08
  • $\begingroup$ So you have something of the form $\exp\{-2t\} \leq 0.5$? $\endgroup$ – jbowman Sep 9 '18 at 18:10

$ P(J) = \frac{e^{-2t}(2)^0}{0!} ≤ 0.5 $

Take Log on both sides:

-2t = -0.69

From here solving for t is easy. The answer should be around 20.79 minutes which you can round to 20 minutes.

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