The density of function X is given by f(x)= a+bx^2 if 0 ≤ x ≤ 1 , 0 otherwise. E(x)=3， find a and b?

Question

vishal_kothari · Answer

http://www.cmpe.boun.edu.tr/courses/cmpe343/fall2009/cmpe343_fall09_ps2_sol.pdf

dumbcow · Answer

$$E(x) = \int\limits_{0}^{1}x*f(x) = \int\limits_{0}^{1}ax+bx^{3}$$
$$\int\limits\limits_{0}^{1}ax+bx^{3} = 3$$
$$\frac{a}{2} + \frac{b}{4} = 3$$
$$b = 12-2a$$
Now the sum of any probability density function is equal to 1
$$\int\limits_{0}^{1}f(x) = 1$$
$$\int\limits_{0}^{1}a+bx^{2} = 1$$
$$a+\frac{b}{3} = 1$$
$$b=3-3a$$
$$\rightarrow 12-2a = 3-3a$$
$$a=-9$$
$$b=30$$

zarkon · Answer

dumbcow's solution is correct given the information provided.  Unfortunately this gives us a density of 

$$f(x)=\left\{\begin{matrix}-9+30x^2,  &  0≤ x ≤ 1 \0, & \text{ow}\end{matrix}\right.$$

and for small values of $x$ we have that $f(x)$ is negative and therefore cannot be a pdf.

dumbcow · Answer

yeah i noticed that and thought maybe something was off