**22.869 probability-theory questions.**

An urn problem
An urn $C$ contains $c\in \mathbb{N}$ elements of three different kinds: There are $\alpha\in\mathbb{N}$ elements of kind $A$, $\beta\in\mathbb{N}$ elements of kind $B$, and $\gamma\in\...

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) \mid \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$
I agree that the probability ...

I came across the following question while studying for a Stochastic Processes exam:
Consider the space $(\Omega,\mathcal F)$, where $\Omega$ is the space of real valued continuous functions, and $\...

Let $\mathcal{Z}\subset \mathbb{R}$ be a set at most countable. Let $X,Y,Z,\xi$ random variables with the following properties:
$X,Y,Z$ have value in $\mathcal{Z}$
$\xi\ $ has a Bernoulli ...

I'm trying to understand how to compute the following expectation.
Let $S$ be exponentially distributed with mean $1$. For any $t > 0$, find, $E[S \mid \max(S,t)]$.
If $S > t$, then $\max(...

Let $\xi_1,\xi_2,\xi_3,\xi_4$ be i.i.d random variable with distribution $N(0,1)$, Find the distribution of
$$ \frac{(\xi_1\xi_2+\xi_3\xi_4)^2}{\xi_2^2 + \xi_4^2} $$
I guass that we can do some ...

Consider $X_{1}, X_{2}, ..., X_{n}$ a sequence of equally distributed independent random variables with density function
$$f(x)= { e^{-x+\theta }}, x \geq \theta $$ \
$$f(x) = 0, x < \theta $$
...

Let $(\Omega, \mathcal{F}, P)$ be a probability space, where $\Omega = [0,1]$, $\mathcal{F}=\mathcal{B}(\Omega)$ is a Borel $\sigma$-algebra and
$P$ is a Lebesgue measure. Given a sequence of random ...

Let $X$ be a binomial random variable with parameter $\left(n,p\right)$, and let $Y$ be a Poisson random variable with parameter $np$. Let $g$ be a convex function. Prove that
$$\mathbb{E}[g(X)] \le \...

Consider the following evolution equation for $c$, given by a convection-diffusion equation over one spatial dimension with positive $D$ and constant $v$:
$c_t = Dc_{xx} - vc_x$
Physical reasoning—...

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...

In Grimmett and Stirzaker's Probability and Random Processes (section 1.3), for two disjoint events $A$ and $B$, we have that
$\mathbb{P} (A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)$
From this ...

Let M be a reversible Markov chain, with stationary distribution Q. Suppose we construct a metropolis Hastings proposal in order to get a chain N with stationary distribution P.
I'd like to bound the ...

Cramér's theorem:
Let $X,Y$ two independent random variables such that $X+Y$ is normal distributed, then $X$ and $Y$ are normal distributed.
I get the the original paper: Harald Cramér. "Über eine ...

I am trying to derive a marginal probability distribution for $y$, and failed, having tried all methods to solve the following integral:
$$p(y)=\int_0^{\frac{1}{\sqrt{2 \pi }}} \frac{\sqrt{\frac{2}{\...

Let $B=(\{\text{heads},\text{tails}\},\mathscr{P}\{\text{heads},\text{tails}\},\mu)$ be the probability space for a single fair coin toss. For any cardinal $\aleph$ let $B^\aleph$ be the independent ...

Let $x$ be a continuous random variable with distribution $D$ and pdf $P(x)$ supported by $\mathbb{R}$. Let $T:\mathbb{R} \to \mathbb{R}$ be an increasing invertible transformation. We denote by $T \...

I want to study the set of solutions of a system of non-linear equations involving a parametric multivariate probability distribution. I would like your help to formalise or contradict some of my ...

I have a set $S$ of $N$ elements; then I do three independent samplings from $S$, and I create three sets $R_1$, $R_2$ and $R_3$ of sizes $N_1$, $N_2$ and $N_3$. Let $N_{int}$ be the size of the ...

Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that
\begin{equation}
\int_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and ...

Let $X_1 \sim \chi_{k}^2$ and $X_2 \sim \chi_{k}^2$ be i.i.d and both $a_1$ and $a_2$ positive real values.
How can be expressed the PDF of $Y = a_1X_1 + a_2X_2$? Is it also a chi-square distribution?...

I need some help understanding a note given in a lot of papers I read.
Let $(\Omega,\mathcal{F},P)$ be a complete probability Space, $\mathbb{F} = (\mathcal{F}_t)_{t\in[0,T]}$ a given filtration with ...

I'm struggling with proving/disproving the following statement.
Let $X$ and $Y$ be two random variables such that $Y \neq g(X)$ and $X \neq g(Y)$ for any function $g$. Then, there are random ...

We arrange $9$ balls numbered $1,\dots,9$ in a row randomly (a permutation). Let $X_i$ an indicator to the ball in the $i$ position is less than the ball in the $i+1$.
Prove $\mathbb{E}[X_2|X_1]=\...

Suppose $\mu$ is a probability measure and $f$ is its characteristic function. Suppose $f$ has finite first order derivative at $t=0$, can you conclude that $\mu$ has finite expectation?
I know that ...

$\newcommand{\P}{\mathbf{P}}$ $\newcommand{\N}{\mathbf{N}}$
I'm trying to solve this problem, and would appreciate a hint on the second part of the problem.
Suppose that $(\Omega, \mathcal{F}, \P)...

Consider two probabilistic events $X$ and $Y$:
if $P(X) = o(1)$ and $P(Y) = \Omega(1)$ (where $o(.)$ and $\Omega(.)$ are little Oh-notation), is it possible to compare the two event $X$ and $Y$? (...

Let $(\Omega,\Sigma,P)$ be a probability space. Suppose that $Y:\Omega \to \mathbb R^n$ and $Z:\Omega \to \mathbb R^m$ are independent, continuous random vectors with probability density functions $...

Let $X_n$ $(n=1,2,\dots)$ be a sequence of discrete random variables, where the distribution of $X_n$ is the discrete uniform over $\{0, 1/n, 2/n,\dots,1 \}$. Let $U$ be a random variable whose ...

I have the following problem:
Let $(M_n)$ be a bounded martingale and $X_n = \sum_{k=1}^{n} 1/k (M_k - M_{k-1})$.
Show that $(X_n)$ is a martingale that converges almost surely and in $\mathcal{L}^2$...

Find b so that f(x) is probability density function.
$f(x)= rect(x-\frac{1}{2})\frac{b}{\sqrt{x}}$
The rules for probability density function are:
f(x) is positive
$\int_{-\infty}^{\infty}f(x)dx=1$
...

In my probability theory lecture the model of a coin toss with a random uniform distributed coin was talked about, which means the probability $(0,1) $ for which a coin shows heads is randomly chosen. ...

If you can show that a random variable is in every $L^p$-space for a given measure, for every $1<p<\infty$ then can you deduce that it is also in $L^1$?

Given 50/50 odds for each chance, what is the probability that I will amass 10 wins before loosing 5-in-a-row.
Downvote for what? I'm looking more for an approach to solving this than the actual ...

I am confused about identifying random variables.
For instance, I have question that says "You roll a die until you roll a 6 a total of four times, and you record the resulting sequence. Which of the ...

Let $X$ be a random variable and $Y=g(X)$
Define
$$\tag{1}
\chi = \{x: f_X(x)>0\}\quad \text{and}\quad \mathcal{Y} = \{y:y=g(x) \text{ for some } x \in \chi\}
$$
Define $g^{-1}(y) = \{x\in \chi:g(...

Let $X$ and $Y$ be mutually independent random variables i-e $$P(X,Y)=P(X) \cdot P(Y)$$ where $P(X)$ is the probability mass function of $X$. Is it possible that some condition (if exists) will make ...

Let $X_n$ be a sequence of uniformly bounded random variables.
If $f(X_1,...,X_n)$ goes to a normal in distribution and $g(X_1,...,X_n)$ goes to a normal in distribution, is it true that $f(X_1,...,...

Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it shows tail, then throw a die. Find the conditional probability of the event that ‘the die shows a number ...

There is a draw. Team H can play against 6 teams (A,B,C,D,E,H) and team B can play against 7 teams (H,J,K,L,M,N,O). What is the probability that there is a match between B and H?

Let $\mathcal L$ be the infinitesimal Generator of a Markov Prozess.
Assume we know, that applied to some suitable, given function $f$ we know that we get the function $a(x)$ i.e
$$\mathcal L f(x)=a(x)...

I need to prove that the distribution of $X=\sum\limits_{n=1}^\infty \frac{X_n}{e^{n}}$ has a Lebesgue measure zero, where $X_n$ is a discrete random variable such that $P(X_n=1)=\frac12$ or $P(X_n=-...

Consider three events $A,B,C$ such that $P(A)>0$, $P(B)>0$, and $P(C)>0$. The events are linked to each other through the constraints $P(A\cup B\cup C)=1$ and $P(A)=P(\overline{B})$. Under ...

Consider three events $A,B,C$ such that $P(A)>0$, $P(B)>0$, and $P(C)>0$. The events are linked to each other through the constraints $P(A\cup B\cup C)=1$ and $P(A)=P(\overline{B})$.
We ...

I have a set of i.i.d random variables $X_i$ with distribution:
\begin{align}
X\stackrel{d}{=}
\begin{cases}
\mathcal U[-z,z] & \text{ if } K_i \leq 0.5 \\
\mathcal U[-1,1] & \text{ if } K_i &...

I post this question with some personal specifications. I hope it does not overlap with old posted questions.
Recently I strongly feel that I have to review the knowledge of measure theory for the ...

If I use the Rao-Blackwell theorem to find that a conditional statistic has the same variance as the original statistic I conditioned on, does that imply that this statistic is a uniformly minimum ...

The goal of this question is to collect standard general facts about convergence of random variables (in $\mathbb L^p$, in probability, in distribution) in order to use them when answering questions. ...

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has (...

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