Solving Distribution Functions

[GreenLRan]

Homework Statement

Given the distribution function:

f(x) = A*sin^2(pi*x/L) for 0 < x < L
= 0 for x elsewhere

Find the values for:
A, mean x (x), most probable x (xmp) and root mean square x (xrms)

Homework Equations

The Attempt at a Solution

The professor has not shown us how to do a problem like this. So I am lost.

My guess at solving A would be to take the integral of f(x) and set it equal to 1. But I do not know what limits I should use on the integral (0 to inf or 0 to L?).

I think the most probable x would be + or - 1 standard deviation from the mean value? I don't know how to solve for that either.

Thanks in advance for the help!

 

[gabbagabbahey]

When in doubt, go back to the definitions.

For starters, for a normalized distribution, $f(x)$ , you have

$$\int_{-\infty}^{\infty}f(x)dx=1$$

In this case, $f(x)$ is zero for most of the interval, and so the only non-zero contribution to this integral comes on the interval $0$ to $L$...What does that give you for $A$?

What are the definitions of "most probable value" and "root mean square x"?...Apply those definitions.

 

[GreenLRan]

Does that mean I should integrate from 0 to L and normalize to 1 to solve for A?

I don't know what the 'definitions' are. Like I said, my professor did not teach this. That is why I am asking for help here. Could you provide a little more insight? Should I integrate x*f(x) to get the most probable value of x?