Independence, the Convolution Formula

Let $\left\{ \xi_{\alpha }, \alpha \in A\right\}$ be an arbitrary collection of independent random variables. Is it true that any countable subcollection $\xi_{\alpha_{1}}, \xi_{\alpha_{2}}, \ldots$, where $\alpha_{1}, \alpha_{2}, \ldots \in A$ are distinct elements, and any Borel sets $B_{1}, B_{2}, \ldots \in \mathscr {B}\left(\mathbb {R}\right)$ satisfy the equality

Let $\xi_{1}, \xi_{2}$ be random variables, each of which is independent of the random variable $\xi$. Is it true that the vector $\left(\xi_{1}, \xi_{2}\right)$ is also independent of the random variable $\xi$?

Let $(\Omega , \mathcal{F}, \mathbb {P})$ be a probability space and $\mathcal{A}$ an algebra of subsets of $\Omega$ such that $\sigma (\mathcal{A})=\mathcal{F}$. Using the principle of appropriate sets, prove that for every $\varepsilon >0$ and $B \in \mathcal{F}$ one can find a set $A \in \mathcal{A}$ such that $\mathbb {P}\left(A \Delta B\right) \leq \varepsilon$.

Let $\mathcal{E}$ be a $\pi$-system of sets. Prove that $\lambda (\mathcal{\varepsilon })=\sigma (\mathcal{\varepsilon })$ (here $\lambda (\mathcal{E})$ is the smallest $\lambda$-system containing $\mathcal{E}$ ).

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables with distribution function $F(x)$. Order the values $\xi_{1}, \ldots , \xi_{n}$ in nondecreasing order. This gives rise to a new sequence of random variables $\xi_{(1)} \leq \ldots \leq \xi_{(n)}$. Find

1. the distribution function of the random variable $\xi_{(k)}$ for each $k \in \left\{ 1, \ldots , n\right\}$

2. the density of the random variable $\xi_{(k)}, k=1, \ldots , n$, if $F(x)$ has density $f(x) ;$

3. the density of the random vector $\left(\xi_{(1)}, \xi_{(n)}\right)$, if $F(x)$ has density $f(x)$;

4. the density of the random vector $\left(\xi_{(1)}, \ldots , \xi_{(n)}\right)$, if $F(x)$ has density $f(x)$.

Random variables $\xi$ and $\eta$ are independent and uniformly distributed on the segment $[0, a]$. Find the densities of the random variables $\xi +\eta , \xi -\eta , \xi \eta , \xi / \eta$.

Write the convolution formula for two discrete random variables, prove it (write the formula both for the discrete density $p_{\xi +\eta }(x)=\mathbb {P}\left(\xi +\eta =x\right)$ and for the distribution function $F_{\xi +\eta }(x)$).

Random variables $\xi_{1}$ and $\xi_{2}$ are independent. Using the convolution formula, find the distribution of $\xi_{1}+\xi_{2}$, if

1. $\xi_{i} \sim \operatorname {Bin}\left(n_{i}, p\right), i=1,2$;

2. $\xi_{i} \sim \operatorname {Pois}\left(\lambda_{i}\right), i=1,2$;

3. $\xi_{i} \sim \mathcal{N}\left(a_{i}, \sigma_{i}^{2}\right), i=1,2$;

4. $\xi_{i} \sim \Gamma \left(\alpha_i, \lambda \right), i=1,2$.

Random variables $\xi_{1}, \ldots , \xi_{n}$ are independent and have the standard normal distribution. Find the distribution of the random variable $\eta =\xi_{1}^{2}+\ldots +\xi_{n}^{2}$.

Let $X, Y$ be independent random variables. Find the probability that a triangle can be formed from segments of lengths $X, Y$ and 1, if

1. $X$ has a uniform distribution on the segment $[0,1]$, and $Y$ is exponential with parameter $1$;

2. $X$ has a uniform distribution on the segment $[0,1]$, and $Y$ has a gamma distribution with parameters $(1,2)$.

A random graph $G(n, p)$ contains $C_{n}^{k}$ collections of $k$ vertices $i_{1}, \ldots , i_{C_{n}^{k}}$. Let $X_{j}=I\left(\left.G(n, p)\right|_{i_{j}}=K_{k}\right), j \in \left\{ 1, \ldots , C_{n}^{k}\right\}$, be the indicator that the induced subgraph of $G(n, p)$ on the collection $i_{j}$ is complete. Is it true that any two random variables $X_{j_{1}}, X_{j_{2}}$ are independent? If not, is it true that for any collection of such random variables, pairwise independence implies mutual independence?

Let $X, Y$ be independent random variables having a uniform distribution on $(0,1]$. Prove that the quantities

are independent and have standard normal distributions.