Density function and expected value

Question

The density function of $X$ is given by: $$f(x)=\left\{\begin{matrix} a+bx^2 & 0\le x\le 1\\ 0 & \text{otherwise} \end{matrix}\right.$$ If $E[X] = \frac{3}{5}$, find $a$ and $b$.

I'm not really sure how to do this problem. Thanks.

Answer 1

You have two equations :

$$\int_0^1 f(x) dx=1,\mathbb{E}[X]=\int_0^1 xf(x)dx=3/5$$

The first one because $f$ must be a density of probability, and the second one is given to you. Express those two integrals in terms of $a$ and $b$ and you will have a system with two equations and two unknowns.

Answer 2

One may recall that, for any density $f$, we have $$ \int_0^\infty f(x)\:dx=1 $$ giving here $$ \int_0^1 (a+bx^2)\:dx=1 $$ or

$$ a+\frac{b}3=1. \tag1 $$

On the other hand, we know that $$ E(X)=\int_0^\infty x \times f(x)\:dx=\frac35 $$ which reads $$ \int_0^1 x(a+bx^2)\:dx=\frac35 $$ or

$$ \frac{a}2+\frac{b}4=\frac35. \tag2 $$

You can then easily conclude.