Find $E[X]$ when $E[aX] = b$ where $E$ is the Expectation.

Question

I know how to calculate the Expectation of random variable. But how to find the $E[X]$ when $E[aX] = b$ where $a$ and $b$ are some positive constants?

Answer 1

There is no $a>0$ such that the value of $Ea^{X}$ determines the value of $EX$. To prove this I will demonstrate that there exist random variables $X$ and $Y$ with $Ea^{X}=Ea^{Y}$ but $EX \neq EY$.

I will leave the trivial case $a=1$ to you.

Let $X=0$ and $Y$ take the values $\pm 1$ with probabilites $\frac 1 {1+a}$ and $\frac a {1+a}$ respectively. Then $Ea^{Y}=a\frac 1 {1+a}+\frac 1 a \frac a {1+a}=1=Ea^{X}$. But $EX=0$ and $EY=\frac 1 {1+a} -(1-\frac 1 {1+a})=\frac {1-a} {1+a} \neq EX$.