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A machine processes parts (one part at a time) with time that is exponentially distributed with rate $\gamma$. Parts are fed to the machine through a conveyor belt according to Poisson process with rate $\lambda$

I am asked to find the probability that $x$ parts arrive at the machine, while the machine is processing a part, given that the machine work on one part in $t$ seconds.

Let $X$ be the number of parts that arrive at the machine during processing. Let $T$ be the time that a machine takes to work on one part.

So, find $\Pr(X = x|T = t)$.

I tried doing something like $\Pr(X = x|T = t) = \dfrac{\Pr(X = x, T = t)}{\Pr(T = t)}... $

However, the correct answer is:

$\Pr(X = x|T = t) = \dfrac{(\lambda t)^x}{x!} \exp(-\lambda t)$

Does anyone see how this solution was arrived?

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During the period $t$ that the machine is processing a part, the expected number of parts to arrive is $\nu=\lambda t$. The probability that $x$ parts arrive is given by the Poisson pmf, $f(x|\nu)$, which is the result you show.

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