Let $(X,Y)$ be a 2d random variable.
Suppose:
$\mathbb{P}(X\in[-1,15])=1$
$\mathbb{E}[Y|X=t]=t^2$
What above could be value for the common $\mathbb{V}$ar of X,Y, which means: $\mathbb{C}_{ov}(X,Y)$
A: $0$
B: $-16$
C: $8$
D: All possible.
To be honest, I really tried this, but I have no idea where to start with it, it is a test exercise and I have no idea how to progress with it.
There was a close question like that, but with $\mathbb{P}(X\in[1,10])=1$ and the solution was $8$.
and another similliar $\mathbb{P}(X\in[-10,-1])=1$ with solution of -16.
again, without knowing why, on both of them
My thoughts regarding the question I asked is maybe D, but I really have no idea why.
Observation: For now, I can give you a partial solution. maybe latter (or with some contribution), I can conclude the final part.
First, consider the following results. Given the Random Variables $U$ and $W$, we have:
R1: (Law of total expectation):$E[ E[U|W]] = E[U]$
R2: $E[h(U)E[W∣U]]=E[h(U)W]$ for every function $h:S \to \mathbb{R}$, where $S$ is the support set of $U$.
Now, we have that $Cov(X,Y) = E[XY] - E[X]E[Y]$.
also, by the problem statement, we have that $E[Y|X = t] = t^2$, which implies that
$$E[Y|X] = X^2 \qquad (1)$$
Applying the R1 over $E[Y|X], we have:
$$E[Y] = E[ E[Y|X] ] = E[X^2] \qquad (2)$$
Applying the R2 with $h$ as the identity function, we have:
$$E[XY] = E[ X E[Y|X] ] = E[X\cdot X^2] = E[X^3] \qquad (3)$$
In that way, we have that:
$$Cov(X,Y) = E[X^3] - E[X^2]E[X] \qquad (3)$$
Now we must investigate the values that Cov(X,Y) could take.
A) Cov(X,Y) = 0.
As pointed in the comments, If $P(X = c) = 1$ for $c \in [-1,15]$, then $E[X^3] - E[X^2]E[X] = (c)^3 - c^2\cdot c = 0$. Therefore, $0$ is a possible value for $Cov(X,Y)$.
This is the uncompleted part!
Until this moment, I could not find any justification to confirm or refute the other possibilities. As soon I have the rest of the solution of this problem, i will update this answer.
reference: https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/04%3A_Expected_Value/4.07%3A_Conditional_Expected_Value