Time interval distribution between events

Question

I am standing in front of a restaurant. Every time a customer goes in the restaurant, I want to make a prediction of dt (dt is the time interval between the next customer and the current customer). What distribution should I use to predict dt? Thanks!

Answer 1

There are a few distributions you can use. You can use an exponential distribution $X\sim \textrm{Exponential}(\lambda)$

$$ f(x;\lambda) =\begin{align}\begin{cases} \lambda e^{-\lambda x} & x \geq 0 \\ \\ 0 & x <0 \end{cases} \end{align} \tag{1}$$

which describes the time between events in a Poisson process. You can also use a Gamma distribution. $X \sim \textrm{Gamma}(\alpha, \beta) $ with mass function

$$ f(x;\alpha, \beta ) = \frac{\beta^{\alpha} x^{\alpha-1} e^{-\beta x} }{\Gamma(\alpha)} \tag{2} $$ where $ x > 0 , \alpha ,\beta >0$

This is an example problem I found.

Example Students arrive at a local bar and restaurant according to an approximate Poisson process at a mean rate of 30 students per hour. What is the probability that the bouncer has to wait more than 3 minutes to card the next student?

Ok, I'm pulling this from here. . It notes if $X$ is the number of students then the poisson mean $\lambda$ we have $30$ per hour or $\frac{1}{2}$ . Denoting $W$ the waiting time between students we would expect the $\theta =\frac{1}{\lambda} = 2$ minutes between every arrival. Since $W$ is exponentially distributed we use the pdf above $ W \sim \textrm{Exponetial}(\frac{1}{2})$

$$ f(w;\frac{1}{2}) =\begin{align}\begin{cases} \frac{1}{2} e^{-\frac{1}{2} w} & w \geq 0 \\ \\ 0 & w <0 \end{cases} \end{align} \tag{3}$$