Determining the normalization constant C

[SirCrayon]

Homework Statement

Consider the distribution function F(x) = Cexp(-ax)

Find the normalization constant C

Homework Equations

The Attempt at a Solution

This is more clarification since this is not actually a homework problem but was in my profs notes. He started with the distribution function above and said we were to determine C.

He then gave us the average equation below:
<z> = ∫ z f(z) dz / ∫ f(z) dz

Which in this case would be xexp(-ax) / exp(-ax)

I am a little confused as to how you are able to determine the constant C if you determine the average of the system/distribution.

 

[vela]

If a distribution is normalized, that means
$$\int f(z)\,dz = 1,$$ where the integral is taken over the entire range of z. In this case, the formula for the average simplifies to
$$\langle z \rangle = \int z f(z)\,dz.$$ The formula your professor gave works for the case where f(z) isn't normalized.

To normalize the F(x) you've been given, you want to find the value of C such that the first integral holds.