# Probability question, exponentially distributed with rate one

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.

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?