Expected value of a single variable Function

Question

$\displaystyle{ f_x(x) = 6x^5 \text{ for } 0 < x <1}$

A.) Find $\mathbb{E}\left[X\right]$

B.) Find $P\bigl[1/3 \le X \le 2/3\bigr]$

Answer 1

Basic probability theory.

A. $E(X)=\int_0^1xf_X(x)dx=\frac{6}{7}$.

B. $P(1/3\le X \le 2/3)=\int_{1/3}^{2/3}f_X(x)dx=(2/3)^6-(1/3)^6$