# Density Functions

• May 15th 2008, 04:54 AM
janvdl
Density Functions
Quote:

A particle of mass $\displaystyle M$ has a random velocity, $\displaystyle V$, which is normally distributed with parameters $\displaystyle \mu = 0$ and $\displaystyle \sigma$. Find the density function of the kinetic energy.
I've asked this before in a previous post, however, while not quite necessary then, the answer did not include full workings, and I think I sort of need that just now to make sure I understand this.
I'm writing another semester test tomorrow, so I would appreciate if someone could show me how this is done.

(Not sure whether it would be classified as "bumping" to bring up the old topic again, I decided to just as well post it in a new thread. Please show full workings.)

• May 15th 2008, 05:23 AM
mr fantastic
Quote:

Originally Posted by janvdl
I've asked this before in a previous post, however, while not quite necessary then, the answer did not include full workings, and I think I sort of need that just now to make sure I understand this.
I'm writing another semester test tomorrow, so I would appreciate if someone could show me how this is done.

(Not sure whether it would be classified as "bumping" to bring up the old topic again, I decided to just as well post it in a new thread. Please show full workings.)

Let $\displaystyle F(x) = \Pr \left( \frac{1}{2} M V^2 < x \right)$

$\displaystyle \Rightarrow F(x) = \Pr \left( -\sqrt{\frac{2x}{M}} < V < \sqrt{\frac{2x}{M}} \right)$

$\displaystyle = \frac{1}{\sigma \sqrt{2 \pi}} \, \int_{-\sqrt{\frac{2x}{M}} }^{\sqrt{\frac{2x}{M}} } e^{-u^2/(2\sigma^2)} \, du$

$\displaystyle = \frac{2}{\sigma \sqrt{2 \pi}} \, \int_{0}^{\sqrt{\frac{2x}{M}} } e^{-u^2/(2 \sigma^2)} \, du$.

Therefore $\displaystyle f(x) = \frac{dF}{dx} = \frac{2}{\sigma \sqrt{2 \pi}} \, e^{-x/(\sigma^2 M)} \, \left (\frac{1}{\sqrt{2Mx}} \right) \,$ , $\displaystyle x > 0$

where the Fundamental Theorem of Calculus and the chain rule have been used to get the derivative

$\displaystyle = \frac{e^{-x/(\sigma^2 M)}}{\sigma \sqrt{\pi M x}} \,$ , $\displaystyle x > 0$

and f(x) = 0 for x < 0.

$\displaystyle \int_{-\infty}^{+\infty} f(x) \, dx = 1$ so I don't think I've slipped up anywhere here .......
• May 15th 2008, 05:25 AM
janvdl
Quote:

Originally Posted by mr fantastic
Let $\displaystyle F(x) = \Pr \left( \frac{1}{2} M V^2 < x \right)$

$\displaystyle \Rightarrow F(x) = \Pr \left( -\sqrt{\frac{2x}{M}} < V < \sqrt{\frac{2x}{M}} \right)$

$\displaystyle = \frac{1}{\sqrt{2 \pi}} \, \int_{-\sqrt{\frac{2x}{M}} }^{\sqrt{\frac{2x}{M}} } e^{-u^2/2} \, du$

$\displaystyle = \frac{2}{\sqrt{2 \pi}} \, \int_{0}^{\sqrt{\frac{2x}{M}} } e^{-u^2/2} \, du$.

Therefore $\displaystyle f(x) = \frac{dF}{dx} = \frac{2}{\sqrt{2 \pi}} \, e^{-x/M} \, \left (\frac{1}{\sqrt{2Mx}} \right) \,$ , $\displaystyle x > 0$
where the Fundamental Theorem of Calculus and the chain rule have been used to get the derivative

$\displaystyle = e^{-x/M} \, \frac{1}{\sqrt{\pi M x}} \,$ , $\displaystyle x > 0$

and f(x) = 0 for x < 0.

Thank you Mr F, I will work through this now and try to understand (Thinking) :D