Statistics Question

Question

At a restaurant, the time a customer has to wait before being seated follows an exponential distribution and is given by the probability density function (pdf):

f(t)=1e−1t

Find the probability that a customer will have to wait at least 6 minutes for a table.

Answer 1

Let X = the number of minutes that a customer waits for a table

As we have an exponential function with pdf given by f(t) = 1e^-1t, we know that X~Exponential(1).

To find P(X ≥ 6), we take the complement, so P(X ≥ 6) = 1 - P(X < 6)

P(X < 6) is the integral from 0 to 6 of (1e^-1t) dt.

Integrating, we have -e^-6 + e^-(0) = 1-e^-6

So P(X≥ 6) = 1-(1-e^-6) = e^-6