# 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.

• The interval between the first random point.... and what? An 'interval' requires two endpoints... – Pierpaolo Vivo Mar 6 '16 at 13:15
• between $t_0$ and the first random point greater than $t_0$ – user3070752 Mar 6 '16 at 13:24

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 $\lambda$.
• I think I found another solution like your solution. If the random variable N denotes the number of random point in the interval $(0,t_0)$ then probability of occurrence of a random point in a given interval is proportional to the interval length so $P(t<X<t+\delta t)=\lambda \delta t$, then we can use some computation to get $P(X<x)=1-e^{-\lambda x}$ – user3070752 Mar 6 '16 at 14:20
• What if $\lambda\delta t>1$? Probabilities do not exceed $1$. – drhab Mar 6 '16 at 14:30
• I think the right expression is $P(t<X<t+\delta t) = \lambda \delta t + O(\delta t)$, when $O(\delta t)$ decrease more quickly than $\delta t$. – user3070752 Mar 6 '16 at 14:39