Q:

How to show a given random variable obey exponential distribution?

I don't know how should I prove a random variable obey a certain distribution. For instance in the following example, how should I start the proof?
Example: if the number of random points on the axis T which are less than $t_0$ obey poisson distribution with parameter  $\lambda t_0$ and the random variable X denote the interval between the first random point on the T axis greater than $t_0$, then X obey exponential distribution.

A:

I hope I understand well.
Event $X>x$ is the event that the number of random points in interval $(t_0,t_0+x]$ equals $0$. 
Probability: $P(N=0)=e^{-\lambda x}$ where $N$ has Poisson distribution with parameter $\lambda$.
So $P(X>x)=e^{-\lambda x}$ making clear that $X$ has exponential distribution with parameter $\lambda$.

