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Currently struggling with this question: Have no idea where to start. This is for my probability course.

Assume that the time between emissions from a radioactive source are independent and exponentially distributed with rate one. Each of these emissions are detected by a Geiger counter with probability p. Compute the density of the distribution of the time between detections of particles, and identify the distribution by name.

Thank you.

Jason Huang
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1 Answers1

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Hint: Let $N$ denote the number of emissions it takes for the Geiger counter to detect an emission. Note that this is a random variable with geometric$(p)$ distribution. Next, let $X_1,X_2,\ldots$ be the sequence of exponentially$(\lambda)$ distributed times between emissions. Then the random variable of interest is $$ X = \sum_{i=1}^{N}X_i, $$ which is gamma$(N,\lambda)$ distributed. Do you know how to proceed from here?

Marc
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  • I'm a little unsure as I cannot find anything on my lectures about this and I struggle with thinking generally. Wouldn't I need a pmf as well? – Jason Huang Apr 22 '17 at 14:46