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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1answer
9 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
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0answers
22 views

If $dX_{t} = X_{t}\,dt + \,dB_{t}$, why does $e^{- t}dX_{t} = e^{-t} X_{t} \,dt + e^{-t} \,dB_{t}$?

I'm taking a course in stochastic differential equations, and in order to solve $dX_{t} = X_{t}\,dt + \,dB_{t}$, the book gives a hint: to multiply both sides of this equation by $e^{-t}$. (But, as ...
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1answer
16 views

Which inequality are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
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13 views

Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
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0answers
24 views

Probability Mass Function of a Sentence

We have a sentence: Some dogs are brown. We choose one letter (out of the 16) at random. Let Y be the length of the whole word containing the letter. How can I find the probability mass function of Y? ...
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1answer
32 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
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1answer
6 views

Fourier transform problem with symmetric matrix. Related to Gaussian?

Hi everyone I encountered a problem that looks simple enough but I have no idea where to start. Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n ...
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0answers
21 views

The “how many pieces do you have buy on average” problem, a markov problem?

I recently discovered a problem similar to this one in a book about Markov chains: Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the ...
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0answers
9 views

Are there different definitions of a continous time Markov chain, condition on a finite or infinite number of earlier values?

My book defines a continous time Markov chain like this. Let $\{X_t\}, t \in T$ be a stochastic process on $(\Omega, \mathcal{A}, P)$, with a countable state space S. The process is a Markov chain, ...
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13 views

Conditional entropy and independent conditioning variables

Let $X,Y,Z,Y',Z'$ be random variables where $Y\sim Y', Z\sim Z'$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are not. Is $H(X|Y,Z)=H(X|Y',Z')$? It seems whether $p(Y,Z)$ factorises or not does ...
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0answers
33 views

If $P(X_1 < X_2)$, what is $P(X_1 < X_2 \cap X_1 < X_3)$?

Say $X_i$ can have a real value in the range [1,100]. All $X_i$ are independent of each other and all values are equally likely. So then $\mathbb{P}(X_1 < X_2) = \frac{1}{2}$, right? But then, ...
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1answer
18 views

uniform Distribution on uncountable Lebesgue $0$-Sets

I know that for every measurable Set A it is possible to create a uniform Distribution on A if - A is finite - A is not a lebesgue 0-set and its not possible for infinite countable sets so I ...
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1answer
14 views

Probability of a vector of normal distribution

Given the set of vectors $\{\mathbf{g}^{1}, \ldots, \mathbf{g}^{N-1} \}$ where $\mathbf{g}^{i} \in \mathbf{R}^M$. Assume that $N \leq M$ and elements of $\mathbf{g}^{i}$ follows normal distribution, ...
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0answers
37 views

Distribution and expectation value of ceiling function of Poisson

There is Poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is less than 1). How can I ...
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0answers
28 views

show that $Y_1$ is unbiased for $\theta$ and find its variance [on hold]

Let $X_1,\ldots,X_n \stackrel {\text{iid}} {\sim} \text{exponential}(θ)$ $$Y_1= \frac {X_1+3X_2+5X_5} {9} $$ $$ Y_2= \sum_{i} X_i$$ Show that $Y_1$ is unbiased for $\theta$ and find its variance. ...
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1answer
23 views

Expected value problem: flip $6$ fair coins before we obtain $3$ heads and $3$ tails?

How many times on average (expected value) must we flip $6$ fair coins before we obtain $3$ heads and $3$ tails? I know I need $∑ xp(x)$. I just don't know how to apply it.
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2answers
21 views

Can covariance (X,Y) be easily expressed in term of Var(X), Var(Y), E(X), and E(Y)?

Can $Cov(X,Y)$ be easily expressed in term of $Var(X), Var(Y), E(X), $ and $E(Y)$ ?
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0answers
10 views

Failure boundary for simple routing problem

As an absolute beginner concerning probability theory I am currently trying to solve the following problem: Given a grid that has $x$ columns (here $x = 4$) and $y$ rows (here $y = 5$), we insert a ...
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0answers
15 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
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1answer
27 views

Poisson Probability with rate $\lambda (t)=-(t-4)^2+16$

The rate at which customer arrive to the bookstore is $\lambda (t)=-(t-4)^2+16 $ where $t$ measured in hours. The customers can buy a book with probability $0.5$ and they can also buy a coffee ...
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1answer
22 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
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3answers
51 views

Upper bound for difference of Poisson random variables

Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then (i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq ...
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2answers
30 views

Coin toss problem. $F_{\alpha} = \{\omega: \frac{\#(k\le n: \omega_{\alpha(k)}=H)}{n}\to \frac{1}{2}\}$

$\Omega=\{H,T\}^\mathbb{N}$, so that a typical point $\omega$ of $\Omega$ is a sequence $\omega=(\omega_1,\omega_2,\dots), \omega_n \in \{H,T\}.$ Let $\mathcal{A}$ be the set of all maps $\alpha ...
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1answer
23 views

Does $\mathbb{P}$-a.s. convergence preserve independence?

Let $\mathcal F$ be a $\sigma$-algebra and $X_n$ RV s.t. $X_n$ is independent of $\mathcal F$ for all $n$. Also let $X_n \to X$ $\mathbb{P}-$a.s.. Is $X$ independent of $\mathcal F$ now too?
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2answers
16 views

Estimate on Probability of a standard normal variable

In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result: If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P ...
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1answer
26 views

Convergence of normal random variables

Let $(\mu_n)_{n \geq 1}$ be a sequence of real numbers and $(\sigma_n)_{n \geq 1}$ be a sequence of positive numbers, and let $\mu \in \mathbb{R}$ and $\sigma>0$. For $n \geq 1$, let ...
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3answers
27 views

Joint probability uniform distribution

I have a question on finding probabilities of joint distributions, specifically two independent random variables that are Uniformly distributed. The question I wish to solve is this one: We agree ...
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3answers
40 views

Largest possible value of $P(A \cap B)$

Suppose $A$ and $B$ are events with $P(A)+P(B)>1$. Show that the largest possible value of $P(A \cap B)$ is $ \min(P(A), P(B))$. I suspect I'm supposed to use $P(A \cap B) = P(A)+P(B) -P(A ...
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1answer
13 views

Pairwise independence implies independence for classes on susbsets of a probability space?

Let $(\mathcal{C}_i)_{i\in I}$ be pairwise independent classes of the probability space $(\Omega, \mathcal{F},P)$, i.e. $\mathcal{C}_i,~\mathcal{C}_j$ are independent for every $i\neq j$ in $I$. True ...
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0answers
12 views

Reduction of Two Independent Random Variables in Quadratic Form

Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
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1answer
22 views

Convergence of product of random variables with distribution $U(0, e)$

Let $X_1, X_2, \ldots$ be a sequence of independent random variables with uniform distribution on $[0, e]$. Let $R_n:=\prod_{k=1}^n X_k$. By Kolmogorov's zero–one law $(R_n)$ converges with ...
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0answers
22 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
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1answer
15 views

Convergence of sequences of random variable

Let $X$ be a random variable. Show that $\frac{X}{n}$ converges to zero in probability and almost surely, as $n \rightarrow \infty$. I am sort of confused by this question since I only learnt a ...
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0answers
13 views

Convexity for Hoeffding's Inequality

We consider a r.v. $X$ that satisfies $0 \leq X \leq 1$ a.s. and a sample of $n$ i.i.d. random variables $X_1,\dots, X_n$ with the same distribution as $X$. We denote by $\mu= E[X]$ and we let ...
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2answers
33 views

$f$ has a zero integral on every measurable set. Prove $f$ is zero almost everywhere

I am trying to solve the following exercise: Let $f$ be integrable. Assume that $\int_A f d\mu = 0$ for every measurable set $A$. Prove that $f = 0$ a.e. [$\mu$]. I have the following proof but it ...
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2answers
237 views

Infinite expectation implies infinite random variable?

Let $X$ be a nonnegative integer-valued random variable depending on a parameter $L$. Assume that $E[X] \rightarrow \infty$ as $L\rightarrow \infty$. Does this imply that for any $k$, $P(X > k)$ ...
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0answers
25 views

Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$.

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
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0answers
23 views

Return time for two independent one dimensional random walks

Let $X^1$ and $X^{-1}$ be two simple random walk in $\mathbb{Z}$ starting respectively from $1$ and $-1$. Let $\tau$ be the first time one of them reaches the origin, $$\tau = \inf \{ j \geq 0 \, : ...
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1answer
33 views

Finding $E[X^{2}]$ of a random variable

I am having a little confusion with finding $E[X^{2}]$ that perhaps can be cleared up relatively easily. Here, $A, B, C$ are Poisson random variables with parameters $2.6, 3,$ and $3.4$, respectively. ...
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16 views

“Mean-field results” in Probability theory

I'm studying a paper on (biological) Neural Networks, and the paper studies some stability properties of an $N$-sized network, and then, as $N$ tends to infinity, it is proven that a "mean-field ...
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19 views

mutual information and data processing inequality for $X\to Y\to Z$ where $Y=f(X)$

Let $X\to Y\to Z$ be three random variables. The data processing inequality states $I(X;Y)\geq I(X;Z)$. Further assume $Y=f(X)$ where $f:\mathcal{X}\to\mathcal{Y}$ is an arbitrary function. What ...
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1answer
125 views

How do I show that the probability of the union of events is not larger than the sum of the individual probabilities?

In numeric analysis class, we are supposed to show that $$P\Bigl(\bigcup_{n\in\mathbb N}A_n\Bigr)\le\sum_{n\in\mathbb N}P(A_n).$$ This is easy to show using induction for a union of finitely many ...
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0answers
31 views

Optimal strategy for a moment to take an exam

In my problem set there was an exercise involving optimal stopping theory. Here is the problem: There is an exam, a list of $n$ questions and $n$ students. Student $A$ knows answers to $k$ of them. He ...
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2answers
29 views

Convergence of Sum of Random variable to another - Cantor function

Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this ...
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0answers
19 views

Sufficient unconditional moment condition for the convergence of $\sum_n (X_n - E[X_n])$

Let $\mathcal F_n$ be a filtration and $X_n$ be $\mathcal F_n$ measurable. Then $M_n = \sum_{k=1}^{n} (X_k -E[X_k])$ is a $\mathcal F_n$ measurable martingale. Let's assume that it is a square ...
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0answers
20 views

Preserving independence of random variables

Suppose I have three random variables, $X,Y,Z$ with $X$ independent of $Z$, $Y$ independent of $Z$. Which transformation can I apply to $X,Y$ to that the result is again a random variable independent ...
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0answers
23 views

SLLN when the expectation in infinite

In a Post I found it says: Whenever ${\rm E}(X)$ exists (finite or infinite), the strong law of large numbers holds. That is, if $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with ...
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1answer
17 views

Absolute expectation of stopped martingale

Let $M_0,M_1,\dots$ be a martingale with respect to $X_0,X_1,\dots$ and $T$ be a stopping time with respect to $X_0,X_1,\dots$ Define $T_n=\min\{n,T\}$ and let $M_{T_n}$ be the stopped martingale. By ...
3
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1answer
47 views

Probability theory with the hyperreals?

Forgive the undoubted ignorance of this question. I am out of my element both in probability and in nonstandard analysis. A mathematically curious layperson friend recently had a conversation with me ...
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1answer
49 views

If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)}$ converges, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$

I have seen in a paper to claim the following: If $\sum\limits_n \sqrt{\mathrm{Var}(X_n)} < \infty$, then $\sum\limits_n (X_n - E[X_n])$ converges in $L^2$, for any sequence of random ...