Solving for c in Probability Density Fucntion

Question

Let X be a random variable with a probability density function (PDF) given by

f(x) = cx^2, if |x|<= 1

          0, otherwise

⦁ Solve for c.

⦁ Calculate E(X)

⦁ Calculate Var(X)

How do we solve this question?

Answer 1

Just use the definitions: $$\begin{align}1~=~&\int_{\lvert x\rvert\leqslant 1} cx^2\operatorname d x & =~& \tfrac 13 cx^3\Big\rvert_{x=-1}^{x=~1}\\[2ex] \mathsf E(X)~=~& \int_{\lvert x\rvert\leqslant 1} x\cdot cx^2\operatorname d x \\[2ex] \mathsf {Var}(X)~=~& \int_{\lvert x\rvert\leqslant 1} x^2\cdot cx^2\operatorname d x~-~\mathsf E(X)^2 \end{align}$$