Conditional distributions and conditional expectations

On the probability space $(\Omega , \mathscr {A}, \mathbb {P})$, where $\Omega = [0,1]$, $\mathscr {A}$ is the $\sigma$-algebra of Borel subsets, and $\mathbb {P}$ is Lebesgue measure, a random variable $\xi$ is given. Let $\mathscr {B}$ be the $\sigma$-algebra generated by the sets $[0,1 / 3),\left\{ 1 / 3\right\}$, and $(1 / 3,1 / 2)$. Find $\mathbb {E}\left[ \xi \mid \mathscr {B} \right]$ if

Under the conditions of the previous problem, find the distribution function of the random variable $\mathbb {E}\left[ \xi \mid \mathscr {B} \right]$.

On the probability space $(\Omega , \mathscr {A}, \mathbb {P})$, where $\Omega = [0,1]$, $\mathscr {A}$ is the $\sigma$-algebra of Borel subsets, and $\mathbb {P}$ is Lebesgue measure, a random variable $\xi = \omega$ is given. Find $\mathbb {E}\left[ \xi \mid \mathscr {B} \right]$ if:

Let $\xi$ and $\eta$ be independent random variables. Find $\operatorname {Var}\left[\mathbb {E}\left[ \xi \eta \mid \xi \right]\right]$ if:

Suppose the random variable $\xi$ takes at most $n$ values. Is it true that $\mathbb {E}\left[ \xi \mid \mathscr {B} \right]$ also takes at most $n$ values?

Prove that if every event of the $\sigma$-algebra $\mathscr {B}$ has probability 0 or 1, then with probability 1

Let $\xi$ be a random variable and $\varphi (x)$ a Borel function such that $\mathbb {E}\left[\varphi (\xi )\right]$ exists. Prove that

Prove that if the $\sigma$-algebras $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ are independent, then for any $\xi$ and $\eta$ the random variables $\mathbb {E}\left[ \xi \mid \mathscr {B}_{1} \right]$ and $\mathbb {E}\left[ \eta \mid \mathscr {B}_{2} \right]$ are independent.

Let $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ be independent $\sigma$-algebras. Prove that for any random variable $\xi$ having an expectation, with probability 1

Let a random variable $\eta = \omega$ be given on the probability space $(\Omega , \mathscr {A}, \mathbb {P})$, where $\Omega$ is the segment $[0,1], \mathscr {A}$ is the $\sigma$-algebra of Borel subsets of $\Omega$, and $\mathbb {P}$ is Lebesgue measure. Prove that for any random variable $\xi$, with probability $1 \quad \mathbb {E}\left[ \xi \mid \eta \right] = \xi$.

Must the random variable $\mathbb {E}\left[ \xi \mid \eta \right]$ be measurable with respect to the $\sigma$-algebra generated by the random variable $\xi$?

Let $\xi$ and $\eta$ be random variables with finite expectation. Prove that if there exists a random variable $\zeta$ such that $\mathbb {E}\left[ \xi \mid \zeta \right] = \eta$, then $\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\eta \right]$. Is the converse true, i.e., does $\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\eta \right]$ imply the existence of $\zeta$ such that $\mathbb {E}\left[ \xi \mid \zeta \right] = \eta$?

Let $\xi$ and $\eta$ be independent identically distributed random variables with finite expectation. Prove that the random variables $\mathbb {E}\left[ \xi \mid \xi +\eta \right]$ and $\mathbb {E}\left[ \eta \mid \xi +\eta \right]$ are identically distributed. In what case are they independent?

Prove that under the conditions of the previous problem

Let $\xi$ and $\eta$ be independent identically distributed random variables with finite expectation. Prove that

Let $\mathscr {B}_{1}, \mathscr {B}_{2}, \ldots$ be a nonincreasing sequence of $\sigma$-algebras, and let $\xi$ be a random variable. Find

Let $\mathscr {B}_{1}, \mathscr {B}_{2}, \ldots$ be a nondecreasing sequence of $\sigma$-algebras, and let $\xi$ be a random variable. Find

Let $\xi$ and $\eta$ be random variables with finite expectations. Prove that if with probability 1 $\mathbb {E}\left[ \xi \mid \eta \right] = \eta$ and $\mathbb {E}\left[ \eta \mid \xi \right] = \xi$, then $\xi = \eta$ a.s.

Let $\xi$ be a random variable with expectation $a$, let $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ be independent $\sigma$-algebras, and let $\xi_{1} = \mathbb {E}\left[ \xi \mid \mathscr {B}_{1} \right], \quad \xi_{2} = \mathbb {E}\left[ \xi_{1} \mid \mathscr {B}_{2} \right]$. Find the distribution of $\xi_{2}$.

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables with finite expectation, and let $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$. Prove that

with probability one.

(Jensen's inequality) Let $\varphi (x)$ be a convex function, and let $\xi$ be a random variable such that $\mathbb {E}\left[\left|\varphi (\xi )\right|\right] < \infty$. Prove that

Prove that $\operatorname {Var}\left[\mathbb {E}\left[ \xi \mid \mathscr {B} \right]\right] \leqslant \operatorname {Var}\left[\xi \right]$.

Let $0 < \alpha \leqslant 1,0 < \beta \leqslant 1, \alpha +\beta \leqslant 1, \xi$ and $\eta$ be random variables with finite expectations. Prove that

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, let $\eta$ be a random variable with finite variance, and let $a = \mathbb {E}\left[\eta \right]$. Prove that

as $n \rightarrow \infty$.

Is the following assertion true: if $\xi_{n} \rightarrow \xi$ a.s., then for any random variable $\eta$

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables, and let $\mathscr {B}$ be a $\sigma$-algebra. Prove that if for some $p \geqslant 1$, $\mathbb {E}\left[\left|\xi_{n}-\xi \right|^{p}\right] \rightarrow 0$ as $n \rightarrow \infty$, then

Let $\mathscr {B}_{1}, \mathscr {B}_{2}, \ldots$ be a sequence of $\sigma$-algebras, and let $\xi$ and $\eta$ be random variables with finite expectations. Prove that if $\mathbb {E}\left[ \xi \mid \mathscr {B}_{n} \right] \rightarrow \eta$ a.s. as $n \rightarrow \infty$, then $\mathbb {E}\left[ \xi \mid \mathscr {B} \right] = \mathbb {E}\left[ \eta \mid \mathscr {B} \right]$ a.s., where $\mathscr {B} = \bigcap_{n = 1}^{\infty } \mathscr {B}_{n}$.

Let $\xi$ and $\eta$ be random variables with finite expectations. Prove that if there exists a sequence of random variables $\zeta_{1}, \zeta_{2}, \ldots$ such that $\mathbb {E}\left[ \xi \mid \zeta_{n} \right] \rightarrow \eta$ a.s. as $n \rightarrow \infty$, then $\mathbb {E}\left[\xi \right] = \mathbb {E}\left[\eta \right]$.

Let $\mathscr {B}_{1} \subset \mathscr {B}_{2} \subset \ldots$ be a nondecreasing sequence of $\sigma$-algebras, and let $\xi$ be a random variable with finite expectation. Prove that for any $\varepsilon > 0$

Let $\mathscr {B}_{1} \subset \mathscr {B}_{2} \subset \ldots$ be a nondecreasing sequence of $\sigma$-algebras,

and let $\xi$ be a random variable with finite expectation. Prove that $\mathbb {E}\left[ \xi \mid \mathscr {B}_{n} \right] \rightarrow \mathbb {E}\left[ \xi \mid \mathscr {B} \right]$ a.s.

Let $\mathscr {B}_{1} \supset \mathscr {B}_{2} \supset \ldots$ be a nonincreasing sequence of $\sigma$-algebras,

and let $\xi$ be a random variable with finite expectation. Prove that $\mathbb {E}\left[ \xi \mid \mathscr {B}_{n} \right] \rightarrow \mathbb {E}\left[ \xi \mid \mathscr {B} \right]$ a.s.

Let $\mathscr {B}_{1}, \mathscr {B}_{2}, \ldots$ be a sequence of $\sigma$-algebras, and let $\xi$ be a random variable with finite expectation. Prove that the sequence $\mathbb {E}\left[ \xi \mid \mathscr {B}_{1} \right], \mathbb {E}\left[ \xi \mid \mathscr {B}_{2} \right], \ldots$ is uniformly integrable.

Prove that the random variable $\xi$ and the $\sigma$-algebra $\mathscr {B}$ are independent if and only if, for every Borel function $\varphi (x)$ such that $\mathbb {E}\left[\left|\varphi (\xi )\right|\right] < \infty$, the equality

holds.

Let $(\Omega , \mathscr {A}, \mathbb {P})$ be a probability space, and let $\mathscr {B}$ be a $\sigma$-algebra, $\mathscr {B} \subset \mathscr {A}$. Prove that for any event $A$

Prove that any two random variables defined on a probability space $(\Omega , \mathscr {A}, P)$ are conditionally independent with respect to the $\sigma$-algebra $\mathscr {A}$.

Prove that the random variables $\xi$ and $\eta$ are independent if and only if they are conditionally independent with respect to the trivial $\sigma$-algebra $\mathscr {B} = \left\{ \varnothing , \Omega \right\}$.

Prove that the $\sigma$-algebras $\mathscr {B}_{1}$ and $\mathscr {B}_{2}$ are conditionally independent with respect to the $\sigma$-algebra $\mathscr {B}$ if and only if for any $B_{2} \in \mathscr {B}_{2}$

Let the random variables $\xi$ and $\eta$ be independent. Can they be conditionally dependent with respect to some $\sigma$-algebra $\mathscr {B}$?

Let $\mathbb {P}\left( A \mid \mathscr {B} \right), A \in \mathscr {A},$ be a regular family of conditional probabilities, and let $\xi$ be a random variable with finite expectation. Prove that

Let $\mathscr {B}$ be the $\sigma$-algebra of Borel subsets of $\Omega = (0,1)$, let P be Lebesgue measure on $\mathscr {B}$, and let $C \subset \Omega$ be a set having outer Lebesgue measure 1 and inner Lebesgue measure 0. Consider the probability space $(\Omega , \mathscr {A}, \mathbf{P}_{c})$, where $\mathscr {A}$ is the $\sigma$-algebra of sets of the form $A = B_{1} C+B_{2} \bar{C}, B_{1}, B_{2} \in \mathscr {B}, \quad \mathbb {P}_{C}\left(A\right) = \frac{1}{2} \mathbb {P}\left(B_{1}\right)+\frac{1}{2} \mathbb {P}\left(B_{2}\right)$. Prove that the family of conditional probabilities $\mathbb {P}_{c}\left( A \mid \mathscr {B} \right)$ is not regular.

Consider the probability space $(\Omega , \mathscr {A}, \mathbb {P})$, where $\Omega = [0,1], \mathscr {A}$ is the $\sigma$-algebra of Borel subsets, and $\mathbb {P}$ is Lebesgue measure. Prove that the family of conditional probabilities $\mathbb {P}\left( A \mid \mathscr {B} \right)$ is regular if:

Let a random variable $\xi (\omega ) = \omega$ be given on the probability space $(\Omega , \mathscr {A}, P)$, where $\Omega = [0,1], \mathscr {A}$ is the $\sigma$-algebra of Borel sets, and $\mathbb {P}$ is Lebesgue measure. Find the conditional distribution of $\xi$ with respect to the $\sigma$-algebra $\mathscr {B}$ generated by the random variable $\eta$, if