# Probability Density

1. Nov 12, 2012

### johnhuntsman

The manager of a fast food restaurant determines that the average time that her customers wait for their food is 2.5 minutes. The manager wants to advertise that anybody who isn't served within a certain number of minutes gets a free hamburger. She doesn't want to give away free hamburgers to more than 2% of her customers. What should the advertisement say?

I'm solving for $t$:

$\int_ 0.4e^{-t/2.5}~dt=0.02$

$-e^{-t/2.5}=0.02$

$0=0.02+e^{-t/2.5}$

Take the natural logs and add $-t/2.5$ to get it back on the other side.

$t/2.5=-3.91$

$t=-1.56$

The answer is ten minutes. What did I do wrong?

Last edited: Nov 12, 2012
2. Nov 12, 2012

### Ray Vickson

You cannot have exp(-t/2.5) = -0.02, since the exponential function is always > 0. You did the integration incorrectly.

RGV

3. Nov 12, 2012

### ehild

You need to show the limits of the integral and calculate with them. The probability that somebody is not served for x minutes is

$$\int _x^\infty{0.4 e^{-0.4 t}dt}=0.02$$

ehild