Can anything be said about the average value of X using its CDF?

Question

Marginal CDF of X:

$$F(x)=\begin{cases} 0 & \text{if }x\in(-\infty,-3)\\ \frac{2x+11}{25}&x \in [-3,2)\\ 1&x\in[2,\infty) \end{cases}$$

Can we say anything about $X$’s average value?

I don’t think we can find the expected value of $X$, because $X$ is not absolutely continuous so we can’t find $X$’s pdf. Also, I said that $X$ is not a non-negative random variable so we can’t do that method either. When I think of saying something about $X$’s average value, I think of the expected value of $X$ or finding a bound of the average value. The latter I don’t think we can do either.

Am I missing something? Can we say anything about X’s average value?

Answer 1

Your rv has 2 discontinuity points

$X=-3$ with probability $\frac{1}{5}$

$X=2$ with probability $\frac{2}{5}$

Thus the expectation is

$$\mathbb{E}[X]=-3\times\frac{1}{5}+\int_{-3}^{2}\frac{2}{25}xdx+2\times\frac{2}{5}=0$$

so we can’t find X’s pdf.

Actually there is the possibility to derive a "mixed PDF" with two discrete points with probability $>0$

This is often used in Telecommunication Enigineering