Probability distribution function proof

[Pomico]

[SOLVED] Probability distribution function proof

Homework Statement

Prove the function p(x) = \frac{3}{4b^{3}}(b^{2}-x^{2}) for -b \leq x \leq +b is a valid probability distribution function.

Homework Equations

I'm not sure if it's as simple as this, but I've been using \int p(x) dx (between b and -b) = 1 if the function is valid

The Attempt at a Solution

\int p(x) dx (between b and -b) = \frac{3x}{4b} - \frac{x^{3}}{12b^{3}} between b and -b = \frac{3}{4} - \frac{1}{12} + \frac{3}{4} - \frac{1}{12} = \frac{4}{3}

I treated b as a constant as I was integrating with respect to x, but clearly \frac{4}{3} is not equal to 1... please help!

 

[HallsofIvy]

You seem to have forgotten the "3" in "3/4" when you multiplied \frac{x^3}{3} by \frac{3}{4b^3}. You should have
\frac{3}{4}- \frac{1}{4}+ \frac{3}{4}- \frac{1}{4}= 1

 

[Pomico]

Heh thanks, stupid mistake...
I assumed it was the integration I got wrong as opposed to an earlier stage, I'll check more thoroughly next time!
Thanks again :)