Can you find an expected value for a random variable that is not non-negative when random variable has a cdf but no pdf?
For instance, say we have that the cdf of X is the following
$$F(x)=\begin{cases} 0 & \text{if }x\in(-\infty,-5)\\ \frac{3x+20}{68}&x \in [-5,3)\\ 1&x\in[3,\infty) \end{cases}$$
X is not a non-negative random variable in this case. We also can't find a pdf of X since X is not absolutely continuous; it has two transition points at x=-5 and x=3.
Do we have any insight into the average value of a random variable that is not non-negative given its CDF?
I was thinking that we wouldn't be able to find the expected value since the random variable is not non-negative and there is no pdf.
I'm sure Greg was assuming you would use the following formula for nonnegative random variables $Z$: $$ E[Z] = \int_0^{\infty} P[Z>z]dz$$ In general you can use $X=X^+-X^-$ and $$ E[X] = E[X^+] - E[X^-]$$ which exists (possibly being $\infty$ or $-\infty$) if and only if the right-hand-side avoids the undefined case $\infty + -\infty$.
If we allow the use of impulse functions, we can often express a PDF for a random variable $X$ in the following form: $$ \boxed{f_X(x) = \sum_{i \in A} w_i \delta(x-x_i) + g(x)\quad \forall x \in \mathbb{R}}$$ where $A$ is a finite or countably infinite set; $\{x_i\}_{i \in A}$ are the points where $X$ has a point mass; $\{w_i\}_{i \in A}$ are the corresponding point masses (so $P[X=x_i]=w_i$); $\delta(x-x_i)$ is a unit impulse function centered at point $x_i$; $g(x) \geq 0$ for all $x \in \mathbb{R}$. In this case we have \begin{align} 1 &= \sum_{i\in A} w_i + \int_{-\infty}^{\infty} g(x)dx\\ E[X] &= \sum_{i \in A} w_ix_i + \int_{-\infty}^{\infty} xg(x)dx\\ E[X^2] &= \sum_{i \in A} w_ix_i^2 + \int_{-\infty}^{\infty} x^2g(x)dx \end{align} If you allow the use of impulse functions, then almost all random variables you will ever see (i.e., random variables that apply to practical problems) have PDFs of the form boxed above. There are some academic counter-examples of random variables that do not have PDFs, even if we allow the use of impulse functions.