# Thread: pdf of random variable

1. ## pdf of random variable

Let $\displaystyle f(x)=\frac{3}{16}x^{2}$, $\displaystyle -c<x<c$

A. Find c so that f(x) is the p.d.f. of a random variable X.
B. Find $\displaystyle \mu$, $\displaystyle \sigma^{2}$ and $\displaystyle \sigma$

Could someone show me how to do this problem, or a problem similar to it? Thank you very much!

2. Hello,
Originally Posted by logitech
Let $\displaystyle f(x)=\frac{3}{16}x^{2}$, $\displaystyle -c<x<c$

A. Find c so that f(x) is the p.d.f. of a random variable X.
B. Find $\displaystyle \mu$, $\displaystyle \sigma^{2}$ and $\displaystyle \sigma$

Could someone show me how to do this problem, or a problem similar to it? Thank you very much!
A.
The pdf of a rv has an integral equal to 1.
So $\displaystyle \int_{-c}^c f(x) ~dx=1$
Calculate this integral and find c

B.
For any bounded and continuous function h, we have : $\displaystyle E(h(X))=\int_{-c}^c h(x)f(x) ~dx$

In particular :
$\displaystyle \mu=E(X)=\int_{-c}^c x f(x) ~dx$

and $\displaystyle \sigma^2=E([X-\mu]^2)=\int_{-c}^c (x-\mu)^2 f(x) ~dx$, which is also $\displaystyle E(X^2)-\mu^2=\int_{-c}^c x^2 f(x) ~dx-\mu^2$

$\displaystyle \sigma=\sqrt{\sigma^2}$, that's all

3. Following your extremely clear example, I found that c=2. Hopefully I calculated correctly in getting $\displaystyle \mu=0$, $\displaystyle \sigma^{2}=\frac{12}{5}$ and $\displaystyle \sigma=\frac{2\sqrt{15}}{5}$

4. Originally Posted by logitech
Hopefully I calculated correctly in getting $\displaystyle \mu=0$, $\displaystyle \sigma^{2}=\frac{12}{5}$ and $\displaystyle \sigma=\frac{2\sqrt{15}}{5}$
Bravo

Following your extremely clear example, I found that c=2.
Actually, this wasn't really an example. It was your exercise

In other problems, you may have different boundaries. The boundaries for the integrals form the region where the pdf is not 0. That's what you have to remember.

Sometimes, you can be given a function f(x)=cg(x), in the interval [a,b] and be asked to find c such that f is a pdf.
In which case, integrate from a to b and find c such that the integral is 1.
Same principle as above, though different question