0
$\begingroup$

Let's throw with a regular dice twice (independently). Let $X_{1}$ be the number of dots thrown on the first try and let $X_{2}$ be on the second. We know, that $X=X_{1} + X_{2}$.

Calculate the expected value $\mathbb{E}[X_{1} \mid X = k ]$.

I'm stuck with this problem, help please.

$\endgroup$
2
$\begingroup$

Well, $k$ can be any resulting sum, so $2 \leq k \leq 12$.

Now if $k = 2$, what could $X_1$ be? Only 1, so $\mathbb{E}[X_1|X=2] = 1$.

Now if $k = 3$, what could $X_1$ be? Either we rolloed $(1,2)$ or $(2,1)$, so $X_1$ can be either $1$ or $2$ with equal probability, so $\mathbb{E}[X_1|X=3] = \frac{1}{2}1 + \frac{1}{2} 2 = 1.5$.

This you can do for any $k$ from 2 to 12...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.