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I have come across a question in my notes and I do not understand how to solve it. I have looked at the solutions and I still am unclear!

This is the question: enter image description here

And this is the solution:

enter image description here

I really don't understand what the question is really asking - would anyone be able to explain to me with an example? I also don't understand what the implication of the conditional probability is: e.g. the prob of 1 goal being scored given that 1 or more goals have already been scored? But then that should be independent?

Sorry - again, I am very confused and any help would be much appreciated!

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  • $\begingroup$ It is more like the probability of exactly 1 goal is scored in total given that 1 or more goals in total are scored so the expected proportion of those games where goals are scored which have exactly one goal $\endgroup$
    – Henry
    Jan 24, 2023 at 13:52

2 Answers 2

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The question has asked, given $N\sim\textrm{Poi}(\lambda), $ to find the value of $\lambda$ such that $\mathbb P[N=1|N\geq 1]= 0.4.$

The solution is nothing but applying the definition of conditional probability and Poisson distribution:

\begin{align}\mathbb P[N=1|N\geq 1]&=\frac{\mathbb P[(N=1)\cap(N\geq 1)]}{\mathbb P[N\geq 1]}\\&= \frac{\mathbb P[N=1]}{\mathbb P[N\geq 1]}\\&= \frac{\frac{e^{-\lambda}\lambda^1}{1!}}{1-\mathbb P[N=0]}\\&=\frac{\lambda e^{-\lambda}}{1-e^{-\lambda}}.\tag 1\label 1\end{align} Equate $\eqref 1$ with $0.4$ and find the value of $\lambda.$

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  • $\begingroup$ Thanks for your answer! Do you mind explaining why P[(N=1) n (N>=1)] = P[N=1]? $\endgroup$
    – Anna
    Jan 24, 2023 at 13:58
  • $\begingroup$ See, Anna, it is the probability of intersection of two events, one being that $N$ is equal to $1$ and other being that $N$ greater than or equal to $1.$ What should be the common event, you think? $\endgroup$ Jan 24, 2023 at 14:05
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    $\begingroup$ (+1) for the clarity! I wonder if it would be useful to also say something about how to solve (1) in $\lambda$... (the OP seem to be confused about it as well ?) $\endgroup$
    – utobi
    Jan 27, 2023 at 18:06
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    $\begingroup$ I could have @utobi but OP didn't respond further. :-) $\endgroup$ Jan 27, 2023 at 22:21
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Conditional probability is a measure of a set (N==1) occupies upon the (N>=1) partition (rather than the whole unity, as it is for unconditional probability). Understanding this fact makes conditional probability much more intuitive.

In a physical sense, among all the cases one or more events of interest happened within a given time interval, 40% of the cases only contained one event. Now, the question is, what was the average event rate per time interval (lambda)?

Next, Poisson is a discrete distribution, so we can calculate the probability of N being 1 using Poisson PMF (lambda ^ 1 * exp(-lambda) / 1!) Probability of N>=1 is also easy to be calculated as 1 - PMF(0).

Calculating via 'trial and error' is of course not the best exercise they could think of but at least it's doable.

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