Assume we have random variables $$X_i \,\,\,\ \text{ i.i.d } \,\,\ i\in[1:n]$$
with expected value $$\mathbb{E}[X_i] = \frac{1}{2}$$
Now let us compute the following expected value of the product of two random variables$$\mathbb{E}[X_1X_2] \stackrel{(a)}{=} \mathbb{E}[X_1]\mathbb{E}[X_2]=\frac{1}{4}$$
If I want to generalize to a product of $n$ random variables
$$\mathbb{E}\left(\prod_{i=1}^n X_i\right)=\prod_{i=1}^n\mathbb{E}\left( X_i\right)= \prod_{i=1}^n \frac{1}{2} $$
so for $n=2$ (to double check with the example above), I have that the expected value of the product is $\frac{1}{2}$ which is wrong, where is the mistake?
Thanks
Best Answer
As a summary of the comments,
$\displaystyle\prod_{i=1}^n \frac{1}{2}$ is the product of $n$ terms each of which is $\dfrac12$, so it is $\left(\dfrac12\right)^n$.
In particular $\displaystyle \prod_{i=1}^2 \frac{1}{2} =\dfrac14$ and not $\dfrac12$.