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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The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma?
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Random Variable Modeling

I am trying to understand how to model a random variable. So using a biased coin with $P(Head) = q$. If I am to generate a random variable $Y$ that is equally likely to be either a or b depending on ...
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How to obtain a certain expression as an expectation

I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a sequence of natural numbers (i.e. this is a probability space of sequence of natural numbers sometimes used in ...
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1answer
20 views

Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$

Question: Let X be a nonnegative random variable and $0 < \lambda \leq EX$. Show that $P(X > \lambda) \geq \frac{(EX - \lambda)^2}{EX^2}$ At first glance I thought I could use some ...
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Showing properties of sigma algebras and algebras

For the first one, my thought is that $\mathcal{A}$ being finite means it has a finite union which is essentially the same thing as a finite countable union. So the result follows. For the second ...
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Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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1answer
19 views

Dyadic expansion

I'm reading the appendix in Billingsley book "Probability and measures" and I can't understand the following. If $$\sum_{i=1}^n\frac{d_i(\omega)}{2^i} < \omega \leq ...
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6 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
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1answer
5 views

Smallest (sub-) Sigma algebra of a null set

Given a probability space ($\Omega,\mathcal{A} ,P$) and $N \in \mathcal{A}, N \ne \emptyset$ with $P(N) = 0$ What is the smallest sub-sigma algebra of $\mathcal{A}$ containing $N$. I'm kind of ...
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Billingsley 2.5a) [on hold]

The field $\mathfrak{F}(\mathcal{D})$ generated by a class $\mathcal{D}$ of subsets of $N$ is defined as the intersection of all fields over $N$ containing $\mathcal{D}$. (a) Show that ...
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argument technique to prove convergence of random variable

I witness a lemma in my class note and I think the proof is not quite clear. Could anybody give me some ideas about argument technique to prove the lemma? The lemma 3 in the beginning of the text: ...
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1answer
13 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...
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1answer
17 views

Why is $P(a < x < b) = P(a < x) - P(x < b)$?

Why is $P(a < x < b) = P(a < x) - P(x < b)$? This is an oversimplified version of a statistics problem I am doing, but I cannot remember why this is true. I know that this works and will ...
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1answer
22 views

Does the Strong Law of Large Numbers imply the following?

The Strong Law of Large Numbers in my Probability textbook is given as follows. Let $X_n$ be a sequence of identically distributed pairwise independent $\mathbb{R}$-valued random variables. Let $S_n$ ...
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1answer
24 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
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12 views

Entropy of the Random Energy Model

I need to show that $$\text{lim}_{N \to \infty}\frac{1}{N}\text{log}\mathcal{N}(\epsilon, \epsilon + \delta) = \text{sup}_{x \in [\epsilon, \epsilon + \delta]}s_a(x).$$ We have that ...
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14 views

Does likelihood ratio has any meaning in this case?

Let $Y$ be a real-valued random variable and $X$ a random variable with realizations in $L^2[0,1]:=L^2([0,1],\lambda)$, where $\lambda$ is a Lebesgue measure. Consider some functional ...
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HOW TO PLOT DAG (DIRECTED ACYCLIC GRAPH) in BNT toolbox for matlab.

I have used markov chain monte carlo (MCMC) in BNT toolbox for matlab, from which i have got one output "sampled_graphs " which is cell array. Now how to plot DAG (Directed acyclic graph ) from ...
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10 views

Equality of two conditional expectations

I would like to show that for any random variable $X$ and $Y$ such that $X$ and $Y$ are independent and for any measurable functions $f$ and $g$, $$ \mathbb E \left[ f(g(X),Z) | g(X) \right] = ...
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2answers
33 views

How do you calculate P(A/B), when event B occurred after event A?

There's really only one question I can't begin to handle when it comes to probability, literally. It's not the only type of question I struggle with, though it's the type of question where I can't ...
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Which properties are stable under convergence in distribution?

I'm currently looking at convergence in distribution (i.e. weak convergence) of random variables. A question is bothering me since quite a while and I hope I could express it properly: Given weak ...
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25 views

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
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11 views

Two random vectors converge does this mean that the entries converge?

Suppose you are given the following two equalities $\mathbf{\delta }^{n}=\left( \begin{array}{ccccccc} \delta _{n,1} & \delta _{n,2} & \cdots & \delta _{n,n} & 1 & 1 & ...
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29 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
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23 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
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4answers
68 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
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12 views

marginal likelihood survivor function [on hold]

Suppose ($x_1$,…,$x_m$) and ($y_1$,…,$y_n$) are drawn from survivor function $S(x;e^{θ/2}) $ and $S(y;e^{-θ/2}) $ respecitively , where $S$ is a defined above. (i) The form of the marginal ...
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17 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
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1answer
21 views

Showing that $\mathcal{A}$ being countable $\Rightarrow f(\mathcal{A})$ is countable - (Algebras/Sigma Algebras)

For the first question my idea was to show that $\sigma(f(\mathcal{A})) \subseteq \sigma(\mathcal{A})$ and $\sigma(\mathcal{A}) \subseteq \sigma(f(\mathcal{A}))$. As for the second question I am at ...
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12 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
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1answer
19 views

Find and sample minimum of two exponential distribtions

I have two (or more) independent exponential variables $ X_1 \sim \exp(\lambda_1) $ and $ X_2 \sim \exp(\lambda_2) $. I want to get both the value of $ \min(X_1, X_2) $ and $ \arg\min(X_1, X_2) $. Can ...
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1answer
36 views

Showing that if $A_{1},A_{2},…$ are all algebras then the union of all of them is an algebra [duplicate]

I am not sure how to show this. It seems obvious but maybe its not. The help would be greatly appreciated!
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1answer
15 views

Is this measure finite, $\sigma$-finite, or a probability measure?

I was a little unsure on this problem. I do have some ideas though. The way I thought of translation invariant is that you can take an interval and shift it, and in the process is will still be the ...
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1answer
48 views

Do not exist IID random variables $X, Y$ such that $X-Y \sim U[-1,1]$

This is an exercise from Williams, Proability with martingales. Prove that if $Z$ has the $U[-1,1]$ distribution, then $$\phi_Z(t) = \frac{\sin t}{t}$$ Then prove that do not exist IID random ...
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1answer
32 views

Playing the St. Petersburg Lottery until I lose everything

This question continues the following question: Calculating the probability of winning at least $128$ dollars in a lottery St. Petersburg Paradox Here is a lottery: A fair coin is flipped repeatedly ...
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50 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
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1answer
43 views

$\frac{S_n}{n} \to -1 \ \ a.e.$, exercise from probability book

I'm stuck with this exercise from Williams, probability with martingales. Let $X_1, X_2, \ldots $be independent random variables with $$P(X_n = n^2-1 )= \frac{1}{n^2}$$ $$P(X_n = -1 )= ...
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2answers
16 views

Value of lambda in poisson distribution

I am currently studying statistical estimators and I came across a question that asks to give an estimate of the parameter λ of a Poisson distribution (using the method of moments), given that the ...
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2 views

Haris recurrent markov chain

What is the definition of haris recurrent markov chain ?. And, when the $\Phi$ irreducibility implies haris recurrent ? Thanks a lot
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1answer
25 views

probability of X+Y which are two independent random variable & uniform distribution[0,1] [duplicate]

Two random variables X, Y are independent and both uniform-distributed in[0, 1]. How to calculate the probability density function Z=X+Y ? I tried below, $$f_X(x) = \begin{cases} \frac1{1-0} \\ ...
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1answer
46 views

Exercise from Williams book Probability with martingales

I'm doing this exercise from Williams book Probability with martingales Let $(X_n)$ be a sequence of IID random variables with $E(|X_n|) = \infty $ for all $n$. Then prove that $$1)\ \sum_n ...
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18 views

Does a state which is passed at least 3 times had to be passed 5 times in Markov chain

Prove of disprove: Let $\{X_n\}_n$ be homogenous Markov chain. if we start from state $i$, there is a positive probability that we pass at least 3 times at state $j$. Does it follows that exists ...
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1answer
26 views

Showing that a collection of intervals (see problem) generates the Borel sigma algebra on $(0,1]$

I would be very appreciative if someone could show me how to do this problem so that I can try to get a better understanding of what a Borel sigma algebra is. Examples are how I learn best so seeing ...
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1answer
32 views

Probability of two elements falling within a certain interval

a, b are selected at random and 0 <= a <= 3 and -2 <= b <=0. What is the probability the distance between a and b is greater than 3. I did this and said that $a$ must be in $(2, 3)$ ...
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1answer
21 views

Applying Ito's formula to a complicated expression

I am faced with some (predictable) process $(r_t)$ and let $0 \leq t \leq T$. I am baffled with the issue of applying Ito's formula to the process $$ \bigg\{ \int_{t}^{T} G(s-t, r_t) \,ds \bigg\}_{t ...
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1answer
22 views

How does a change of measure affect covariance?

Suppose I have the three random variables $X,Y,M$ where $E[M] = 1$ under the measure $P$. Now, suppose I define a new measure $\widetilde P$ so that $\widetilde E[X] = E[M X]$ and $\widetilde E[Y] = ...
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12 views

local martingales and dividend processes

Consider a $d+1$ asset, continuous-time model where asset $0$ is a riskless numeraire. Assume that the asset prices are modelled by a $(d+1)-$dimensional Ito process (B,S). Further, let $D$ be the ...
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1answer
24 views

How do I prove that a given probability distribution is Gaussian

I am trying to plot the distribution of a random variable $x$. I got this distribution by marginalising a wishart distribution. When I plot the distribution curve of $x$, it looks like bell shaped ...
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15 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
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2answers
39 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...