Given a PMF, find $E[\frac 1x]$?

Question

Ok, so this is homework and I've been able to do ALMOST all of this problem but I need a push in the right direction for this part.

Suppose that the random variable $X$ has pmf $f(x) = \frac 1{25}x$ for x $\in$ $\{1, 3, 5, 7, 9\}$.

I've gotten most of the parts of this problem correct but one of them asks you to find $E[\frac 1x]$. I have no idea how to do this. I've been researching for a while now but have as of yet come up with nothing to show for it. I know that you can go from $E[aX + b]$ to $aE[x] + b$, but I have no idea how I'm supposed to do this one.

Answer 1

Note that $$ E(g(X)) = \sum_{x \in \{1,3,5,7,9\}} g(x)P_X(x), $$ hence in your case you have $$ E(1/X) = \frac{1}{25}\left(1 \times\frac{1}{1} + 3 \times\frac{1}{3} + \cdots + 9 \times\frac{1}{9}\right) = 1/5. $$ Or more quickly $$ E(1/X)=\sum \frac{1}{x}\times\frac{x}{25} = 5\times\frac{1}{25}=1/5. $$