Expectation of $1 - x^{0.5}$

Question

Let $x$ be an indicator variable such that $E[X] = \frac{1}{3}$, calculate $E[1-x^{0.5}]$.

I'm having a hard time figuring out why this isn't equivalent to $E[1] - E[x^{0.5}]$ which would be $1-(\frac{1}{3})^{0.5}$

any help would be greatly appreciated.

Answer 1

I'm not sure what you mean by an indicator variable. If you mean a binary (0/1) variable then the answer is that $\sqrt{x}=x$ and hence $E[1-\sqrt{X}]= 1- EX=2/3$.

More generally, please note that $E[X^a]\neq (E[X])^a$ except in very special cases like when $a=1$ or $X$ is constant.

Answer 2

Hint: Because $X$ is an indicator variable, what is $ P (X = 0)$?

Hint: The linearity of expectation applies to the sum and product, but not to the composition of functions. In general,

$$ E[X^a] \neq E[X]^a.$$

Hint: Because $X$ is an indicator variable, hence $ E[X] = E[X^{0.5}]$.