what does E[$c^x$] mean in probability

Question

Hello I'm self studying probability this summer and I would like your help to clarify me on this question.

Let x be such that $P(X=1)=p=1-P(X=-1)$.

Find $c≠1$ such that $E[c^x]=1$

Can anyone tell me what does $E[c^x]$ mean in probability?

I know that $E[x]$ is the expected value and it's a number but $E[c^x]$ ???

Answer 1

EDIT: In general, for any function $f(x)$, we have $$ E[f(x)]=\sum_{i}f(i)P(x=i). $$ END EDIT

Expanding on the comment above, $$ E[c^x]=c^1P(x=1)+c^{-1}P(x=-1)=pc+\frac{1-p}{c}. $$ Thus if we have $E[c^x]=1$, we want $$ pc+\frac{1-p}{c}=1 $$ or $$ pc^2+(1-p)=c $$ or $$ pc^2-c+(1-p)=0. $$ The quadratic formula can now be used to solve for $c$ in terms of $p$.