Bound on variance

Question

I would like some help with my homework assignment.

I do not expect a full solution but i do not know where to even start.

How do i even start with part a

Answer 1

How to show (a):

You have $P(\{X\geq b\})=\beta$

Note that $x^2$ is a increasing function when $x\geq 0$ and $b>0$. Then $\inf \{x^2:b\leq x<\infty\}=b^2$

$$\int_b^\infty x^2f_X(x)\,dx\geq b^2\int_b^\infty f_X(x)\,dx=b^2P(\{X\geq b\}) =b^2\beta$$

Answer 2

\begin{align} \int_b^\infty x^2 f_X(x)\,dx & \ge \int_b^\infty b^2 f_X(x)\,dx & & \text{because } x^2\ge b^2 \text{ when } x\ge b >0 \\[10pt] & = b^2\beta & & \text{by the definition you gave for } \beta. \end{align}