Exponential distribution

In this post, I explain the exponential distribution and its relationship to Poisson distribution.

Motivation: Assume that at a car wash station, on average 6 cars arrive every hour. We are interested in the probability that the time until the next car arrives is $x$ minutes.

We can look at this problem as follows: the event that the time until the next car arrives is $x$ minutes, is the same as the event that there are no cars arriving between now and the next x minutes, i.e. the interval $[0,t]$. This is the Poisson distribution with parameter $\lambda t$.

$$\begin{align} P(X = 0) &= \frac{(\lambda t)^x e^{-\lambda t}}{x!} \\ &= \frac{(\lambda t)^0 e^{-\lambda t}}{0!} \\ &= e^{-\lambda t} \end{align}$$

$P(X=0) = e^{-\lambda t}$ can also be interpreted as the probability that the time, $T$, until the first event is greater than $t$, i.e.

$$P(T > t) = P(x = 0 | \mu = \lambda t) = e^{-\lambda t}$$

Thus the probability that the next car arrives within $t$ minutes is

$$P(T < t) = 1 - P(T > t) = 1 - e^{-\lambda t}$$

This is the cumulative distribution of $T$. Differentiating gives the pmf $f(t) = \lambda e^{-\lambda t}$.