I am working on the problem of finding the cumulative distribution function of a variable $X$, which can take values $-1$, $0$, or $1$, has expectation $0.1$, and variance $0.89$.
I don't understand how to go about this problem, as all the examples I can find online go the other way. From what I understand, I need to find three separate probabilities, one for each value $X$ can take, which should add up to $1$, correct? How do I go about doing this?
Yes, that is the right approach. If you let the probabilities of $-1,0,1$ be $a,b,c$ you have three equations in three unknowns. One is $a+b+c=1$ because the sum of the probabilities has to be $1$. One comes from the expectation, which is $(-1)a+(0)b+(1)c=0.1$. The third comes from the variance. Write the variance in terms of $a,b,c$ and solve the equations simultaneously.