Minimize expected cost with early and late penalties for appointment

[twoski]

Homework Statement

Suppose that if you are s minutes early for an appointment, then you incur cost s * $3,
while if you are s minutes late, you incur cost s * $5. Suppose the travel time from your
present location and the location of the appointment is a continuous random variable with
pdf f(x) such that f(x) = (1/10)e^-x/10 if x ≥ 0 and f(x) = 0 if x < 0. How many minutes before your appointment should you depart in order to minimize your expected cost?

The Attempt at a Solution

So i want to find E[X] and then the variance i assume...

$E[X] = \int_{0}^{∞} x( 1/10e^{-x/10})$

But if i evaluate this I'm going to get either 0 or infinity... So should this just be an indefinite integral?

 

[Ray Vickson]

Homework Statement

Suppose that if you are s minutes early for an appointment, then you incur cost s * $3,
while if you are s minutes late, you incur cost s * $5. Suppose the travel time from your
present location and the location of the appointment is a continuous random variable with
pdf f(x) such that f(x) = (1/10)e^-x/10 if x ≥ 0 and f(x) = 0 if x < 0. How many minutes before your appointment should you depart in order to minimize your expected cost?

The Attempt at a Solution

So i want to find E[X] and then the variance i assume...

$E[X] = \int_{0}^{∞} x( 1/10e^{-x/10})$

But if i evaluate this I'm going to get either 0 or infinity... So should this just be an indefinite integral?

Computing EX has nothing to do with the problem. You want to look at expected COST, which will depend on when you leave. Write a formula for the cost function, given that you leave m minutes before your appointment and the trip takes x minutes. Then take the expected value of that cost function.