Question

Find E(X), Var(X), SD(X) if a random variable X is given by its density function f(x), such that:
f(x) = 0, if x <=0
f(x)=3/8x^2, if 0 < x <=2
f(x)=0, if x>2

Answer 1

\[E(X)=\int_{-\infty}^\infty x f(x)~dx=\int_0^2\frac{3}{8}x^3~dx\]
\[V(X)=E(X^2)-(E(X))^2=\int_{-\infty}^\infty x^2f(x)~dx-(E(X))^2=\cdots\]
Finally, \(SD(X)=\sqrt{V(X)}\).