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Raindrops hit you at a rate or 1 raindrop per second, what is the PDF of the time at which the second raindrop hits you?

Clearly, we have to use exponential random variable. Also we are asked to use convolution.

In this case, I think we can split the situation to: $X$ and $Y$ in which $X$ is the first drop and $Y$ is the second drop. And to use convolution, we have to know the PDF of both $X$ and $Y$.

For $X$, it would be $1/e$, but for $Y$, I am not sure.

Can we say that PDF of $Y$ is equal to PDF of $X$ plus $1/e$? Since the PDF of second drop could be thought of the same scenario as $X$ but with some initial time added for the first drop to fall?

I am a bit confused with the usage of exponential random variables here and how we can transform this into convolution. I would appreciate some help.

Thanks.

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