Conditional Distributions and Computing Conditional Expectations

Let $X, Y, Z$ be independent identically distributed random variables. Find $\mathbb {E}\left[4X-3Y+Z\mid X+Y+Z\right]$.

Let the random variables $X$ and $Y$ be independent, with the density of $X$ equal to $\frac1x I_{[1,e]}(x)$, and the density of $Y$ equal to $4x^3 I_{[0,1]}(x)$. Compute $\mathbb {E}\left[\frac1X+\frac1Y \mid XY\right]$.

Let $X$ and $Y$ be independent random variables, uniformly distributed on the interval $[0,2]$. Find $\mathbb {E}\left[Y^2 \mid X/Y\right]$.

Let $X$ and $Y$ be independent random variables, $X$ having a uniform distribution on the interval $[0,1]$, and $Y$ an exponential distribution with parameter 1. Find (a) $\mathbb {E}\left[Y \mid X/Y\right]$, (b) $\mathbb {E}\left[Y^3/X^2 \mid X/Y\right]$.

Let $X_1,\dots ,X_n$ be a sample from the distribution $R(0,1)$. Find (a) $\mathbb {E}\left[X_1\mid X_{(1)}\right]$, (b) $\mathbb {E}\left[X_1\mid X_{(n)}\right]$.

Let $X_1,\dots ,X_n$ be a sample from the distribution $R(0,1)$. Find (a) $\mathbb {E}\left[X_1\mid X_{(1)}\right]$, (b) $\mathbb {E}\left[X_1\mid X_{(n)}\right]$.

Find $\mathbb {E}\left[3X^2 + 5X - 5XY - 2Y^2 - 10Y \mid 2Y-X\right]$ given that $X$ and $Y$ are independent, $X$ has the standard Laplace distribution (with parameter $\sigma =1$), $Y\sim R(-1,2)$.

Let $\xi_1,\xi_2,\dots$ be independent identically distributed random variables with finite expectation, $S_n=\xi_1+\dots +\xi_n$, $n\in \mathbb {N}$. Prove that for all $n\in \mathbb {N}$ the equality $\mathbb {E}\left[\xi_1\mid S_n, S_{n+1}, \dots \right] = \frac{S_n}{n}$ (a.s.) holds.