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 [tag:probability-...

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49 views

Convergence of the number of visits in a Markov Chain

Suppose we have an irreducible and recurrent discrete-time Markov chain with states over the finite set $\mathcal{X}$. Let $N_t (x)$ denote the number of visits to state $x$ up to time $t$. Let $\pi(x)...
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0answers
27 views

Summation methods and entropy

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} =...
0
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1answer
29 views

Find the density function of the random variable $Z=X_1+X_2$

Problem: Let $X_1,X_2$ be independent variables with common density function $f(x)=0.5e^{-0.5x},0<x<\infty.$ Find the density function of the random variable $Z=(X_1+X_2)$. My attempt: Let $...
3
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1answer
151 views

Bound the variance of the product of two random varables.

For two random variables $X$ and $Y$ show that the following inequality holds $$\mathrm{Var}(XY)\leq 2\|Y\|_{\infty}^{2}\mathrm{Var}(X)+2\|X\|_{\infty}^{2}\mathrm{Var}(Y).$$ Well first I tried to ...
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3answers
57 views

Determine the Density Function of $(X_1/X_2)$

Problem: Let $X_1$ and $X_2$ be independent and uniformly distributed random variables over (0,2). Determine the Density Function of $(X_1/X_2)$. Here are my thoughts on the problem: $\vec{x}=(x_1,...
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1answer
53 views

An independent squence of functions that are uniform on $[0,1]$

Suppose that $X$ is uniform in $[0,1]$. Find an infinite sequence of functions $f_{i}$ so that all $f_{i}(X)$ are independent and uniform $[0,1]$. um I'm not really sure how to do this. I'm thinking ...
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1answer
53 views

Proof for k-connectedness of random graphs

I am really new to the theory of random graphs. It seems a lot of articles take for granted that: For $k\in\mathbb{N}\setminus\{0\}$ and $p\in(0,1)$ fixed, almost every graph in $G(n,p)$ is $k$-...
3
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1answer
80 views

If $P$ a probability of a sentence to be true, then $\{P(\phi | T_i)\}_{i \in \mathbb{N}}$ is a martingale over constructed theories $T_i$

I am reading Section 2.1 of Definability of Truth in Probabilistic Logic. For a language $L$, fix a probability distribution $P:L \to [0,1]$. Enumerate sentences $\phi_1, \phi_2, \ldots$ of a ...
4
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0answers
76 views

When will this generalized binomial model generate an exchangeable sequence?

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ $$P(X_1 = 1)= \theta_0$...
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1answer
27 views

Finding a measurable function with an independent uniform distribution

Suppose $X,Y,U$ are random variables on some probability space such that $U$ is independent of $(X,Y)$. Prove there exists a measurable function $f: \mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ ...
4
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1answer
137 views

Simplest Solution for a Round Table Q

Company of 3 Turks, 3 British and 3 French sit at a round table. What is the probability that no two countrymans sitting next to each other? All the people are different, but sitting orders different ...
3
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1answer
58 views

Series of independent gaussian variables and brownian motion

I am checking the proof of the construction of a brownian motion in $[0,\pi]$. We show that \begin{gather*} t \mapsto B^m_t = \frac{t}{\sqrt{\pi}}X_0 + \sqrt{\frac{2}{\pi}}\sum_{n=1}^{2^m-1}X_n \frac{\...
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2answers
68 views

Frechet-Hausdorff theorem reference from J.L. Kelley used in proof that each probability measure is inner regular

Theorem: If $S$ is a complete, separable metric space, then each probability measure on it is inner regular. Proof: Since $S$ is separable, for each $n \in \mathbb{N}$ there exist countably many ...
5
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0answers
49 views

Generalized Binomial Model independent in the limit

Start with a generalized binomial model $$P(X_{n+1}=1\mid \mathcal{F}_n)=\theta_n+ n^{-1} d_n \sum_{i=1}^n X_i$$ $$P(X_{n+1}=1)=p_{n+1}=\theta_n + n^{-1}d_n \sum_{i=1}^n p_i$$ With $0\leq \theta_n+ ...
0
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1answer
45 views

Random variable related to binomial

The number of successes $A$ in $n$ independent trials with the probability of a success is $p$ for each trial is binomially-distributed. I am interested in a scenario that adds dependence to the ...
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1answer
33 views

Show that the σ-algebras generated by the collection of all intervals are equivalent

Show that the σ-algebras generated by the collection of all intervals of the form [a,b]⊂R and by the collection of all the intervals of the form (−∞,b]⊂R are equivalent. i am having trouble with ...
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1answer
65 views

Sum of two random variable X and -X

Let $X$ and $Y$ be two random variables such that $X$ and $-Y$ have the same distribution. Prove that $P(X+Y<-t)=P(X+Y>t)$. I know how to prove this when $X$ and $Y$ are independent but how do I ...
0
votes
1answer
70 views

Martingale $X_n \to \infty$ a.s.

Construct a martingale $X_n$ such that $X_n \to \infty $ a.s. I have trouble coming up with such an example and prove it. Can someone provide an example?
4
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0answers
448 views

Adapted but not progressively measurable?

Let $X(t,\omega)$ be a stochastic process: $$ X: \mathbb{R}^+ \times \Omega \rightarrow \mathbb{R}, $$ where $(\Omega, \mathcal{F}_t, \mathcal{F}, \mu)$ is a stochastic basis. Some definitions: $...
5
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1answer
432 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times [0,1]$....
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1answer
87 views

characteristic function upper bound and uniformly continuous.

Let $X$ be a random variable and let $\phi$ be its characteristic function. Let $A$ be a nonnegative constant and consider the following inequality $$ |\phi(t)-\phi(s)| \leq \sqrt{A|1-\phi(t-s)|}. $$ ...
2
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1answer
39 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
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2answers
761 views

Covariance of uniform distribution and it's square

I have $X$ ~ $U(-1,1)$ and $Y = X^2$ random variables, I need to calculate their covariance. My calculations are: $$ Cov(X,Y) = Cov(X,X^2) = E((X-E(X))(X^2-E(X^2))) = E(X X^2) = E(X^3) = 0 $$ because $...
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2answers
4k views

Find the probability density function of $Y=X^2$

Consider the random variable X with probability density function $$f(x)=3x^2$$ if $0<x<1$, and $$f(x)=0$$ otherwise. Find the probability density function of $Y=X^2$. This is the first question ...
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0answers
89 views

Reference requests for an opt-cited result in Jennrich (1969)

Lemma 2 on page 637 of Jennrich (1967) states that: Let $Q$ be a real-valued function on $\Theta\times Y$ where $\Theta$ is a compact subset of a Euclidean space and $Y$ is a measurable space. ...
0
votes
1answer
35 views

A basic question on the existence of expectation?

$E\big[\sqrt(X)\big] <\infty \implies \sqrt(X) <\infty$ a.s $\implies X< \infty$ a.s $ \implies E[X] <\infty$ The expectation is computed wrt to the probability measure . So why the the ...
1
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1answer
68 views

expected value of three uncorrelated random variables

Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven? Note: we don't know if they are ...
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1answer
31 views

Tightness of random variales

If $\{X_n\}$ is a tight family of positive r.v.s. can we say something about $\{f(X_n)\}$ where $f$ is a continuous function?
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0answers
292 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; i\...
1
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1answer
78 views

A generalization of the Glivenko-Cantelli theorem

Let $P$ and $P_n$ be probability measures on $\mathcal{B}(\mathbb{R})$ with distribution functions $F$ and $F_n$. Moreover, let $F$ be continuous and $(P_n)_{n\in\mathbb{N}}$ weakly converge to $P$. ...
4
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1answer
306 views

Intuition behind Strassen's theorem

Currently I am dissecting a proof of Strassen's theorem, which states the following: Suppose that $(X,d)$ is a separable metric space and that $\alpha,\beta>0$. If $\mathbb{P}$ and $\mathbb{Q}...
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0answers
75 views

Let $X_n>0$ be iid and $P(X_n>t)\sim t^{-\alpha}$, show that $Y_n=n^{-1/\alpha}S_n$ and $1/Y_n$ are tight.

We are given that $X_n>0$ be iid with common distribtuon $X$, and $P(X>t)\sim t^{-\alpha}$, I need to show that the scale of $Y_n$ is $n^{1/\alpha}$. Or in other words show that $Y_n=n^{-1/\...
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2answers
67 views

Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of $X(t)$,...
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0answers
92 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
0
votes
1answer
46 views

If $P(A_n) \ge \epsilon>0$ for large $n$, then $P(A_n i.o.) \ge \epsilon$

If $P(A_n) \ge \epsilon>0$ for large $n$, then $P(A_n i.o.) \ge \epsilon$. I tried mimicking the proof of B-C but it give the wrong inequity in a different direction.
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0answers
98 views

Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
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2answers
411 views

Prove that if $X$ is stochastically larger than $Y$ then $E(X)\ge E(Y)$

Prove that if $X$ is stochastically larger than $Y$ (i.e. $P(X > t) \ge P(Y > t)$ then $E(X)\ge E(Y)$. I understand how to solve the problem if $X$ and $Y$ are non-negative random ...
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votes
1answer
84 views

Let $A$ be the set of irrational numbers in $[0,1]$. Show that $P(A)=1$

Let $A$ be the set of irrational numbers in $[0,1]$. Show that $P(A)=1$ , where $P$ is Lebesgue measure. What ever we do there are infinite irrational numbers for every two rational numbers, right? ...
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0answers
47 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
0
votes
1answer
60 views

Finiteness of the number of big jumps of a Lévy process on a finite interval

Let $Y(t)$, $0 \leq t \leq 1$, be a Lévy process. Denote by $\{ \Delta Y_i \}$ the jumps of the process. I would like to show that the set $\{i: |\Delta Y_i| > r\}$ is finite almost surely. However,...
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2answers
173 views

Show that Y=aX+b is an random variable. [closed]

Let X be an random variable on a given probability space and let a,b∈R. Show that Y=aX+b is an random variable. if X has a distribution function F, what is the distribution function of Y? if X ...
2
votes
1answer
348 views

Almost sure convergence using Borel-Cantelli lemma

Let $(X_n)$ be a sequence of random variables. I want to show that if $E[X_n] \rightarrow C$ and $Var(X_n) \leq \frac{C}{n^2}$, where $C$ is some constant, then $X_n$ converge almost surely to $C$. I ...
0
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0answers
36 views

What is the limiting distribution of this Markov Chain?

Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$: Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the ...
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1answer
124 views

Question about probability in infinite game of chance

Imagine a game where you start with \$100 and toss a coin repeatedly. If it's heads - you lose \$1, if its tails - you double your money. Game ends when you lose all the money. Given infinite amount ...
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0answers
40 views

Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = \...
2
votes
1answer
75 views

Mean of Poisson distribution

Let $X$ have a Poisson distribution with double mode at $x=1$ and $x=2$. Find $ P(X=0)$.Here is my solution: $$\mu= \frac {p(2) 2!}{p(1)}$$ Then how can find the mean? Thanks.
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0answers
39 views

What is the transformation that maps a Gaussian distribution to a Beta distribution?

Suppose X is a random variable with Gaussian distribution over domain $\mathbb{R} = (-\infty, +\infty)$, with PDF function $f_X$. And Y is a random variable with Beta distribution over domain $[0,1]$,...
0
votes
1answer
50 views

Equivalence of $\sigma$-algebras: generated by $[a,b]$ and $(-\infty,b]$

Show that the $\sigma$-algebras generated by the collection of all intervals of the form $[a,b]\subset\Bbb R$ and by the collection of all the intervals of the form $(-\infty,b]\subset\Bbb R$ are ...
6
votes
1answer
1k views

Confusions about Radon-Nikodym derivative and dominating measures

I have some difficulties to understand the Radon-Nikodym derivative and link it to the ordinary way of obtaining the probability density function, which is through the derivative of cumulative ...
2
votes
1answer
27 views

Martingale Conceptual Question

For a normal random walk where $Y_i = \pm\frac{1}{2}$ with equal probability and $X_i = \sum_{i=1}^n Y_i$, my book says the $\sigma$-algebra generated by a martingale is written as $\sigma(X_0, X_1, ...