Continuous random variable from Poisson distribution

Question

X, which represents the number of emission from a radioactive substance in n seconds, follows a Poisson distribution with a mean of 3n.

The first part of the question asks for an expression for the probability that there are no emissions in a period of n seconds. I was able to solve this by using the definition of the Poisson and reached an answer of e^-3n. Part B defines a continuous random variable that represents the 'time until next emission'. The books says that the previous answers can be used to derive the PDF, but I can see no way of doing that. Any Help?

BTW: The answer keys says that the solution is 3e^-3n.

Answer 1

Let $Y$ be the time, in seconds, until the next emission. By your calculation, $\Pr(Y\gt y)=e^{-3y}$ (for $y\gt 0$). Thus for $y\gt 0$ the cdf $F_Y(y)$ of $Y$ is given by $$F_Y(y)=\Pr(Y\le y)=1-e^{-3y}.$$ Differentiate. The density function $f_Y(y)$ of $Y$ is given by $f_Y(y)=3e^{-3y}$ (for $y\gt 0$). Thus $Y$ has exponential distribution with parameter $3$.