The number of passenger planes that crash every day follows the Poisson distribution with parameter p. The number of crashes each day is independent. What is the probability of exactly 3 planes crashing in 8 days?

I think this exercise is somewhat unclear since it doesn't give enough information. Is there a solution to it, because I can't think of anything. So, far I've used the Poisson distribution mass probability and I didn't reached any conclusions.

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    $\begingroup$ You should get a result wich depends on $p$. $\endgroup$ – AlexR Apr 27 '15 at 12:28
  • $\begingroup$ @AlexR Can you be a little more specific? I've tried using the mass probability ,but I can't reach a definitive number $\endgroup$ – Marios Ath Apr 27 '15 at 12:31
  • $\begingroup$ Can you do it if p = 0.3? If you can, replace the 0.3 with the symbol p. This is what @AlexR means. $\endgroup$ – Paul Apr 27 '15 at 12:59
  • $\begingroup$ @Paul Thanks Paul. However, I don't think I am allowed to replace p with an independent number. $\endgroup$ – Marios Ath Apr 27 '15 at 13:01
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    $\begingroup$ If $X$ is the number of crashes in $1$ day and has distribution $Poisson(p)$ and if $Y$ is the number of crashes in $8$ days then $Y$ has distribution $Poisson(8p)$. $\endgroup$ – Mick A Apr 27 '15 at 14:27

Perhaps use of the letter $p$ for the Poisson average daily rate is confusing you. This is not a probability. Many books use $\lambda$ (lower-case Greek 'lambda') for this rate.

You may not be able to replace $p$ with a number in your final answer, but that may be a worthwhile suggestion to get started understanding the problem. Suppose the average rate per day is 0.3 as in one of the comments. Then the average rate per 8 days is 8(0.3) = 2.4. (Notice that the rate 2.4 > 1 cannot possibly be a probability.)

Then let $X_8 \sim Pois(2.4)$ and use the Poisson formula to find $P(X_8 = 3) = e^{-2.4}(2.4)^3/3!.$ I'm guessing that formula may be in your textbook or notes.

Now that you see how it works for an average daily crash rate of 0.3, you should be able to write $P(X_8 = 3)$ in terms of the general symbol $p$ (or $\lambda$ or whatever).

In any Poisson problem you have to make sure that you use the average rate that applies to the question at hand. That may or may not be the rate given at the start of the problem. For example, here we had to adjust the given rate of p per day to the rate 8p per eight days in order to answer a question about what happens over eight days.


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