1. ## Compute E[X]

Compute E[X] if X has a probability density function given by
$\displaystyle f(x) = \begin{cases} c(1-x^2) & -1<x<1 \\ 0 & otherwise \end{cases}$

---------------------------------My work ---------------------------------

$\displaystyle E[X] = \int_{-1}^{1} x. f(x) dx = \int_{-1}^{1} x . [c (1-x^2)]dx = ......=0$.........Is this correct????

2. Hi zorro, first find c by saying

$\displaystyle \int_{-1}^{1} c(1-x^2) dx = 1$

$\displaystyle c\int_{-1}^{1} (1-x^2) dx = 1$

To make things easier you can use some symmetry

$\displaystyle 2c\int_{0}^{1} (1-x^2) dx = 1$

3. i am getting $\displaystyle c = \frac{3}{4}$............Is that right????

what should i do now??????

4. Originally Posted by zorro
i am getting $\displaystyle c = \frac{3}{4}$............Is that right????

what should i do now??????
Now calculate $\displaystyle \frac{3}{4}\int_{-1}^{1} x (1 - x^2) \, dx = \frac{3}{4} \int_{-1}^{1} x - x^3 \, dx$. Since the integrand is odd, the integral is equal to zero (which is what you had in the first place).

Note that the value of c is irrelevant for this particular calculation. But it might be needed for other calculations ....

5. thanks my fantastic and pickslides
cheers