How to solve this propability density function?

Question

I have a function, which I think is a exponential probability function. It is defined as this:

$$f(x) = \begin{cases} \frac{c}{x^3}, & \text{if $x$ $\geq$ 1} \\[2ex] 0, & \text{else} \end{cases} $$

Now, the question is: Determine the value of $c$, such that $f(x)$ is indeed a proper density function. And: Determine $E(X)$ and $P(X>EX)$.

PS: I have tried to solve the question by using properties of the exponential function, but was confused by determining the $c$.

Thank you in advance for explaining the situation for me.

Answer 1

The probability mass must be 1, so $$\int_1^\infty c/x^3 =c/2$$ yields the equation: $$c/2=1$$ Assuming that the domain for $x$ is all of the real line.