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### Intuitively, why does the conditional expectation of X given the trivial sigma algebra equals the expected value of X itself?

Let $F_0=\text{{$\emptyset, \Omega$}}$ be the trivial sigma algebra for a probability space $(\Omega, F, P)$. Let $X$ be some integrable random variable defined on $(\Omega, F, P)$. Intuition that can ...

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### Is it true that $E[X|Y] = \rho \frac{\sigma_X}{\sigma_Y} Y$ if $X$ and $Y$ are not jointly gaussian?

Let $X,Z$ be iid standard Gaussian. Let $Y= |Z| \, sgn(X)$. This forces $X$ and $Y$ to have same sign; in terms of the joint density function, it amounts to erasing two quadrants and multiply by two ...
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If $Y$ and $Z$ are conditionally independent when given $X$, then \begin{align}\mathsf P_{X,Y\mid Z=z}(x,y\mid z)=\mathsf P_{Y\mid X=x}(y)\,\mathsf P_{X\mid Z=z}(x)\\\mathsf P_{Y\mid Z=z_0}\!(E)=\... • 129k 1 vote ### Calculating \mathbb{E}[2\sin (\pi Z)|\cos (\pi Z)] when Z is uniform on [0,2] I have an upcoming exam, so I wanted to tackle this question as an exercise. We use the second approach suggested by the accepted answer. Let X=\cos(\pi Z) and Y=\sin(\pi Z) where Z \sim \text{... • 149 1 vote ### Expected number of die rolls to get 6 given that all rolls are even. The probability of rolling a 6 is \frac16 and of a 2 or 4 is \frac26. The conditional expectation is therefore\dfrac{1\times \frac16+2 \times \frac16\left(\frac26\right)^1 +3 \times \frac16\...
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