Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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4
votes
1answer
260 views

Partial sum of numbers

My TA gave today this question as a nice question to think about. He said its involves standard ideas of Probability theory and numbers. But, I don't even know how to start. Let $x_1, \ldots, x_n$ ...
2
votes
1answer
93 views

Do two probability kernels induce the same distributions if they induce the same distributions on rays?

If two probability kernels induce the same distributions when evaluated on sets of the form $\left(-\infty,t\right]$, do they induce the same distributions when evaluated on all Borel sets? More ...
1
vote
1answer
178 views

Almost Sure Convergence in $L^{p}$

Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. random variables, on the same probability space, with law given by $\displaystyle \mathbb P(X_1=(-1)^{m}m)=\frac{1}{(cm^2\log m)}$ for $m\geq 2$ where $c$ ...
2
votes
1answer
42 views

Independence of a function and integral of a function

I have a probability space $(\Omega, M, P)$ and non-negative integrable functions on $\Omega \times [0,1]$, $F_1(\omega, t)$ and $F_2(\omega, t)$. For each $t \in [0,1]$, we have that $F_1(\omega, t)$ ...
2
votes
1answer
84 views

flipping a coin repeatedly

A biased coin $P(H) = p$ (and $P(T) = q$) is tossed repeatedly, let X be the length of the first run and Y be the length of the second run. A run is a maximal sequence of consecutive heads or ...
1
vote
1answer
112 views

Probability Question - Conditional Probability

Hi I'm working with particle filters at the moment however my maths isnt so strong i was wondering given the $P(X_n|Y_0,....Y_{n-1})$ and $P(Y_n|X_n)$ how do you obtain $P(Y_n|Y_0,...,Y_{n-1})$? i.e. ...
6
votes
3answers
345 views

Proof that $E(X)<\infty$ entails $\lim_{n\to\infty}n\Pr(X\ge n) = 0$?

As the title says. I think this should follow straightforwardly but I can't find a proof. My random variable of interest $X$ takes values in the non-negative integers. The only other assumption on ...
0
votes
1answer
71 views

Find the distribution of a random variable

Let $\Omega=[0,1]$, $\mathcal{F}=\mathcal{B} \cap [0,1]$, and $P$ be the Lebesgue measure restricted to $[0,1]$. Let $\Phi_{\mu,\sigma^2}(x)=\mathcal{N}_{\mu,\sigma^2}((-\infty,x]) $. Then it is clear ...
2
votes
0answers
20 views

Measurability of a certain function

I have a probability space $(\Omega, M, P)$ and non-negative integrable functions on $\Omega \times [0,1]$, $F_1(\omega, t)$. Let $X_1(\omega) = \int_a^b F_1(t,\omega) \ dt$, $0 \leq a< b \leq ...
1
vote
2answers
57 views

expectation of products…

I'm trying to build $m$ real random variables $X_1,\dots,X_m$ such that $$\mathbb{E}[\prod_{i\in\alpha}X_i]=0\quad\forall\alpha\subsetneq\{1,\dots,m\}\,,$$ $$\mathbb{E}[\prod_{i=1}^mX_i]\neq 0\,.$$ I ...
0
votes
1answer
549 views

Uniform sum distribution

I was wondering how to derive the probability density function for the sum of $n$ independent iid distributed random variables on the interval $[0,1]$. A formula for that is given on ...
3
votes
0answers
62 views

Convex Hull of Sampled Random Points

Consider $N$ i.i.d. random points $x_1,x_2,..., x_N\in \mathbb{R}^n$, sampled from a given distribution $d$ defined on $\mathbb{R}^n$. Let $\mathcal{C}_x \subset \mathbb{R}^n$ be the convex hull of ...
0
votes
2answers
168 views

$E(X\mid X\lt x)$ with $X\sim\text{Exp}(a)$

$X\sim\text{Exp}(a)$. How do I calculate $E(X\mid X\lt x)$? Workings: \begin{align}E(X|X<x)&=\int_0^txf(x|x<x)dx\\ &=\frac{\int_0^txP(x,x<x)dx}{\int_0^tP(x<x)dx}\\ ...
0
votes
1answer
94 views

How can I approximate numerically the $\operatorname{Erf}(x)$ function using the Fresnel integrals?

I know that $$C(Z)+iS(Z)=(\pi/2)^{1/2}\cdot\frac{1+i}2\cdot\operatorname{Erf}(z)$$ but I do not know how to go from here, because what I want to approximate is the real value of ...
2
votes
1answer
71 views

Expectation of squared random variable

Let $W$ be defined as $W=(\sum_{i=1}^{n}V_{i})^{2}$ and $\left|V_{i}\right|\leq 1$. Determine that $\mathbb{E}[\left|W\right|]<\infty$. What I have so far updated version: ...
3
votes
1answer
154 views

successive doubling the stake until head appears

I consider the following gaming system: Start with 1 dollar and always bet on head (coin tossing). You always double your stake until the first head appears. Maximum rounds: $n$ I formulated it as a ...
1
vote
2answers
515 views

Definition of atomic $\sigma$-field.

Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
0
votes
3answers
51 views

How to deal with probability questions of this type?

First of all, I would like to say that I still didn´t have a proper probability class (only some introductory class that dealt more with statistics than with pure probability) that would cover this ...
3
votes
1answer
58 views

Measure of a set satisfying $ x \in B$ if and only if $x + k/2^n \in B $

Say we have a probability space $(\Omega, M, P) = ([0,1], \text{ Borel Sets}, \text{ Lebesgue measure})$. Suppose we have the following set $B \in M$ such that $x \in B$ if and only if $x + k/2^n \in ...
0
votes
2answers
93 views

Probability problem: about a radioactive substance

A substance undergoes radioactive decay by emitting particles at random times. Observation shows that the probability of an emission is proportional to the observation time, in the limit of small ...
1
vote
1answer
63 views

Conditional independence and factorization

Does $a \perp\hspace{-1.3ex}\perp b \mid d$ imply $p(a,b,c\mid d) = p(a,c\mid d)\ p(b,c\mid d)$?
0
votes
1answer
61 views

Question involving Martingale

I have a probability space $(\Omega, M, P)$ and non-negative integrable functions on $\Omega \times [0,1]$, $F_1(\omega, t)$ and $F_2(\omega, t)$. For each $t \in [0,1]$ we have that $F_1(\omega, t)$ ...
0
votes
1answer
49 views

On independence

I have a probability space $(\Omega, M, P)$ and non-negative integrable functions on $\Omega \times [0,1]$, $F_1(\omega, t)$ and $F_2(\omega, t)$. For each $t \in [0,1]$ we have that $F_1(\omega, t)$ ...
2
votes
1answer
395 views

Condition Expectation of Difference between Two Poisson processes

$P_t$ and $Q_t$ are poisson processes with rates $a$ and $b$. How do I calculate $E[(P_t-Q_t)]^2|Q_t=m-P_t]$?
1
vote
1answer
166 views

Conditional variance of arrival times

Given a poisson process $P_t$ with rate $r$, with arrival times $S_n$ How do I calculate the Variance of $S_2-S_1|P_t=2$?
0
votes
1answer
387 views

Conditional CDF of Poisson process

$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$ (independent processes) $n = 1,2,3...$ Find the conditional CDF $F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$ I get an answer of ...
4
votes
2answers
456 views

Distribution of infinite sum of Poisson distributed r.v.

Let $X_1,X_2\dots$ all be independent, Poisson distributed with parameter $l_i$ each. Then it is known that for each n $S_n:=\sum_{i=1}^n X_i\sim \text{po}(\lambda_n)$ where $\lambda_n:=\sum_{i=1}^n ...
0
votes
1answer
176 views

Is there another solution for write the algorithm by matlab?

I have an algorithm as you can see below and i wrote it with different 2 ways but it seems there are problems for both solutions.is there any alternative to write it with different way? $$X_i(k+1)= ...
0
votes
1answer
69 views

An equivalent condition for integrability of a measurable function?

Did said that for any nonnegative random variable $Z$, $E (Z \times I_{Z > z}) \to 0$ as $z \to \infty$ is equivalent to $Z$ being integrable, i.e. $E Z < \infty$. Here is my proof for one ...
6
votes
1answer
468 views

Martingale not uniformly integrable

I've come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can't think of an ...
1
vote
1answer
87 views

Paths on $\mathbb{Z}^d$

Let's say a path must be non-self-intersecting, and that we have the usual lattice structure. Then if $\sigma(n)$ is the number of paths of length $n$ then why do we have convergence of the sequence ...
3
votes
1answer
80 views

Lipschitz Continuity of Optimal Value

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $m(W) = 1$. Consider a function $f : X \times Y \times W \rightarrow [0,1]$, where $X \subset \mathbb{R}^n$, and $Y \subset ...
1
vote
1answer
82 views

How does Lindeberg CLT imply classicial CLT?

For a sequence of zero-mean $\sigma^2$-variance IID random variables $X_1, \dots,$ the Linderberg condition is applied this way: $\forall \epsilon >0$, as $n$ goes to $\infty$, $$\sum_{i=1}^n ...
2
votes
1answer
42 views

Proof of theorem involving a positive RV X

I'm trying to understand the proof of the following theorem: If $X \geq 0$ is a random variable, then $$ \int_0^\infty x\ dF_x(x) = \int_0^\infty [1 - F_X(y)]\ dy. $$ The proof is as follows: $$ ...
2
votes
2answers
402 views

Stopping time proof

Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider $$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process ...
2
votes
1answer
453 views

RV's X,Y uniformly distributed in the 2D shape — More information?

In my study of probability I've come across exercise questions talking about random variables uniformly distributed in disks and triangles. I'd like to know if there are "standard" ways of ...
4
votes
2answers
425 views

Kolmogorov's maximal inequality and convergence of random series.

Let $(X_n)_{n\ge 1}$ be a sequence of mutually independent random variables, on the same probability space, with expectation 0 and finite variance. Let $S_n = \sum_{l=1}^n X_l$. Prove that for any ...
2
votes
1answer
64 views

Hitting time level and Bernoullis

Let $(X_n)_{n\ge 1}$ be a sequence of i.i.d. Bernoulli random variables, on the same probability space, with parameter $1/2$, and let $\tau_n$ be the hitting time of level n by the partial sums, i.e. ...
1
vote
1answer
423 views

Partition induced from sigma-algebra in Sigma itself?

Let $(\Omega, \Sigma)$ be a measurable space. Define $$P(\omega)=\bigcap_{\omega \in F \in \Sigma}F$$ P() induces a equivalence relation: $\forall \omega \in \Omega, \omega \in P(\omega) $. $\forall ...
3
votes
1answer
128 views

Asymptotics of $\max\limits_{1\leqslant k\leqslant n}X_k/n$

I found an assertion in this paper at the beginning of page 6, but i can't see how to justify it: Let $X_n \geq 0$ i.i.d. with finite expectation then: $$ \frac1n\max\limits_{k \leq n}X_k \to 0 ...
2
votes
1answer
44 views

Let $X^n\sim Y^n$ for all $n$ and $X^n\to X$ and $Y^n \to Y$ both in probability. Is $X\sim Y$?

Let $X^n\sim Y^n$ for all $n$ and $X^n\to X$ and $Y^n \to Y$ both in probability. Is $X\sim Y$? If all variables take values in some measurable space $(S,\mathbb S)$: I'm thinking $P(X\in A)=\lim \, ...
4
votes
2answers
663 views

Covariance of two random variables with monotone transformation

Suppose I know that, for two random variables $X,Y$, we have $$Cov(X,Y)\neq 0.$$ What happens if we take a monotone transformation of $X$; will the inequality persist? That is, say $f(.)$ is a ...
2
votes
2answers
626 views

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece

We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece
0
votes
1answer
186 views

Does Not Converge in Probability?

Let $\left(X_n\right)_{n\geq 1}$ be a sequence of i.i.d. real random variables, with $\mathbb E(X_1)=0$, var$(X_1)=1$. Let $S_n=X_1+\cdots+X_n$. Prove that $\displaystyle ...
1
vote
1answer
56 views

Independence of $n$ random variables

Let $A_1,A_2,\ldots,A_n$ be independent subsets of probability space $(\Omega, \Sigma, P)$ (For every $I\subseteq \{1,2,\ldots,n\}$, $P(\bigcap_{j\in J}A_j)=\prod_{j\in J}P(A_j) )$. Prove that ...
2
votes
0answers
63 views

Consider a bipartite graph and Deduce that there exist infinitely many bipartite graphs

Consider a bipartite graph $G=(X \cup Y, E)$ with both sides of equal size; we let $n$ denote $|X|=|Y|$ We are also given an integer $d\ge 3$ and we wish each vertex in $X$ to be adjacent to at most ...
1
vote
1answer
52 views

$r^{th}$ factorial of Hypergeometric Distribution

$r^{th}$ factorial $E(X_r)=E[X(X-1)\dots(X-r+1)]$ in case of Hyper-geometric Distribution. $$f_X(x) = \left\{ \begin{array}{ll} \frac{\binom{a}{x}\binom{N-a}{N-x}}{\binom{N}{n}} & x ...
2
votes
0answers
130 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...
1
vote
1answer
111 views

estimation of a moment for the sum with Bernoulli random variables

Let $x\in R_+^n$ and let $b_i, i=1, \ldots, n$ be $(0,1)$ Bernoulli random variables with $P(b_i=1)=p$. Denote $S=\sum_{i=1}^n x_ib_i$. For $q\geq 2$ estimate from above $$ E\left|S\right|^q $$
1
vote
1answer
108 views

cdf induced by a cdf

Let $(E,\Sigma, \mathbb{P})$ be a probability space, and $X: E\to \mathbb{R}$ a random variable. $F$ is the cdf of $X$. Define a new random variable $Y:=F(X)$. What is the cdf of $Y$?