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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116 views

Covariance matrix of a Brownian motion

Suppose that $Y$ is a d-dimentional brownian motion under a setting $(\Omega, \mathbb{F}, P)$ adapted to a filtration ${F_t}$. Then is the covariance matrix of $Y$ always diagonal? In other words is ...
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1answer
22 views

Conditional expectation of integral

$$E\Big(\int_0^2 t^2W_t^3 \, dt \mid F_1\Big)=\int_0^1 t^2W_t^3 \, dt +\int_1^2 E(t^2W_t^3 \mid F_1) \, dt=$$ $E(W_t^3\mid F_1)=E((W_t-W_1+W_1)^3\mid F_1)=E((W_t-W_1)^3\mid ...
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1answer
51 views

Conditional probabilty calculation

I have two probabilities: P(A) = 0.002 ... The probability that someone has a specific disease P(B) = 0.995 ... The ...
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2answers
134 views

The Hurried Duelers brainteaser

This question is similar as this other one asked in the forum, but I am trying to give it a different twist. Unfortunately, I am not getting to the same answer, so there might be something wrong in my ...
2
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1answer
108 views

Show that a process is gaussian

I need an help with the following exercise. I would like to know if what I've done is correct. Let $(X_t)_{t\geq 0}$ be the process define as $$X_t=e^{\lambda t} X_0-\sigma \Big(\lambda \int_0^t ...
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1answer
23 views

New measure and expectation formula

If we define new measure to be $Q(A) = \int _AZ(\omega)dP(w)$, then $E_Q(X)=E_P(XZ)$. But why this euqality is true? Is there any formal way of showing it? Does $dQ(w)=Z(\omega)dP(\omega)$ hold? Why? ...
3
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1answer
193 views

Does convergence in probability imply a.s. convergence in a countable space?

Let $(\Omega, \mathcal F,\mathbb P)$ be such that $\Omega$ is countable. I'm trying to find a simple example of random variables $X_n$ which converge to $0$ in probability but not a.s. If $\mathcal F ...
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1answer
118 views

What is an example of pairwise independent random variables which are not independent?

I've just read in a stochastics textbook: Let $(\Omega, P)$ be a discrete probability space. (a) The events $A_i \subseteq \Omega, i=1,2, \dots$ are called independent, if $$P(A_{i_1} \cap ...
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0answers
57 views

Find minimum variance & C.R.L.B?

Let $X_1$, $X_2$, $X_3$, ... , $X_n$ be a random sample from a distribution having mean µ and variance $σ^2$. Also, Let $Y_1$, $Y_2$, $Y_3$, ... , $Y_m$ be a second random sample from the same ...
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1answer
83 views

Conditional expectation acting on square of a random variable

Suppose that $X$ and $Y$ are random variables such that $E(Y¦X) =X$ and $E(Y^2¦X)=X^2$; also, $Y$ is in $L^2(\Omega,\mathcal{A},\mathbb{P})$. I need to show that $Y=X$ almost surely. I know the ...
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1answer
70 views

Brownian motion and posterior distribution

I am a bit stuck on this question: Suppose that $X_t = W_t + \alpha t$, where $W$ is a standard Brownian motion, and let $\mathcal{F}_t = \sigma ( X_u: 0 \leq u \leq t)$. The drift is constant in ...
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1answer
65 views

Covariance in normal lognormal (NLN) mixture

Let $u = \epsilon e^{\frac{1}{2} \eta}$ where \begin{equation*} \left( \begin{array}{c} \epsilon \\ \eta \\ \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \\ ...
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2answers
327 views

Condition implying tightness of sequence of probability measures

A sequence of probability measures $\mu_n$ is said to be tight if for each $\epsilon$ there exists a finite interval $(a,b]$ such that $\mu((a,b])>1-\epsilon$ For all $n$. With this information, ...
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0answers
17 views

A simpler way for an entropy inequality

I have to show that $\frac {1}{N}H(X_1,...,X_N)\le H(X_1)$. for a stationary stochastic process. I know that $H(X_1,...,X_N)=\Sigma _{i=1}^N H(X_i|X_1,...,X_{i-1})$. So far I have plugged that ...
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0answers
37 views

Ito formula of a product

I would like to calculate stochastic differential of: $$X_t=\left(\int_0^t(s^3+B_s) \,dB_s \right)(2t+tB_t)=Y_tZ_t$$ I would like to use: $d(Y_tZ_t)=Z_t \, dY_t +Y_t \, dZ_t+dY_t \, dZ_t\tag{$*$}$ ...
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61 views

Distribution of the quotient of two random variables

Let $x$ and $y$ be two random variables with support of $\left[1\hspace{5pt}10\right]$ and $\left[50\hspace{5pt}90\right]$ respectively. The distribution of each of these variables is $p_X(x)$ and ...
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1answer
76 views

An immortal crazy hen

An immortal hen was put in an empty house. Every day she lays $2$ eggs, and every evening she breaks $1$ unbroken egg, chosen uniformly at random from the remaining unbroken eggs. What is the ...
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0answers
32 views

Is $\sigma(X_1X_2)\subseteq\sigma(X_1,X_2)$ true?

Given two real valued random variables $X_1$ and $X_2$, I think that $\sigma(X_1X_2)\subseteq\sigma(X_1,X_2)$, but I can't prove it. Here $\sigma(X)$ denotes the $\sigma$-algebra generated by the ...
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1answer
53 views

Probability on $C(\mathbb{R})$

Let $C(\mathbb{R})$ be the set of continuous and bounded functions $\mathbb{R}\to\mathbb{R}$. Is there a probability measure $p$ on $C(\mathbb{R})$ such $\forall g\in C(\mathbb{R}),\ \forall ...
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0answers
52 views

subtracting mean of iid RVs increases mutual information?

I have a problem about intuition: substracting the mean of iid RVs seems to increase the mutual information. Say $X,Y$ are real iid RVs, then $\frac{X-Y}{2}$ and $\frac{Y-X}{2}$ are not independent ...
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1answer
54 views

Gärtner-Ellis theorem reference

Could you please suggest a good reference for the Gärtner-Ellis theorem (at a rigorous, postgraduate student level)? Many thanks for your help.
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56 views

Why is the probability that two independently sampled values have the same value is zero?

In a book about machine learning, it reads, Generally, the probability that $x$ generated independently by a continuous probability distribution $p(x)$ have the same value is zero. Otherwise, ...
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1answer
55 views

Terminology: Probability “with respect to a measure”

The following excerpt is taken from Shen and Wasserman (2001). I have difficulty understanding some terminologies. On line 4, [...] each $P_\eta$ is a probability on $(\mathscr Y,\mathscr ...
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2answers
37 views

Flip biased coin ind. Find recursion $p_n$ of event of at least $3$ consecutive $H$ in $n$ tosses, in terms of $p_{n−1}, p_{n−2}, p_{n−3}$. Find $p_6$

I know the recursion formula. It is $p_n = p_{n-1}q + p_{n-2}pq + p_{n-3}p^2q + p^3$, but I am having trouble finding $p_6$. I was thinking it would be $p_3 = p^3$ since it would have to be $H$ on ...
3
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1answer
237 views

The expected time until reaching a specified set in a Markov chain

I am reading an article in which they discuss a specific Markov chain in an example, and it turns out I need to sharpen up my Markov knowledge. First the setup. I have a continuous time Markov chain ...
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2answers
106 views

Proving Cantelli's inequality

I'm assuming that the random variable $X$ has mean $0$ and finite variance ${\sigma}^2$. It is immediate from Chebyshev's inequality that $$P(X\geq x)\leq \frac{{\sigma}^2}{x^2},$$ but I'm still ...
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0answers
47 views

Can an arbitrary probability space be simulated by coin tosses?

Let $B=(\{0,1\},\mathscr{F},\mu)$ be the probability space for a single coin toss, and for any cardinal $\aleph$ let $B^\aleph$ be the product probability space. For each probability space ...
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2answers
134 views

Probability of infinite intersection.

I came to the following problem: Let $A_1, A_2, ...$ be events in a probability space $(\Omega, F, \mathbb{P})$ and $\mathbb{P}[A_j]=1$ for all $j>1$. I need to show that the probability of the ...
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1answer
91 views

Using Lindeberg’s Condition together with the Central Limit Theorem

I have the following problem: Problem. Let $ (X_{n})_{n \in \mathbb{N}} $ be a sequence of independent random variables such that $$ \mathbf{Pr} \! \left( X_{n} = \sqrt{n} + 1 \right) = ...
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2answers
34 views

Variance of the compound sum

Trying to solve a Variance evaluation problem: Now I'm not sure how to evaluate those two terms on the right hand side of the last equality... Would appreciate any help.
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1answer
94 views

Prove that process is uniformly integrable

Let $(X_t)_{t\ge 0}$ be a stochastic process, and let $Y$ be an integrable random variable, such that $|X_t|\le Y$ for $t\ge0$. Prove that $(X_t)_{t\ge 0}$ is uniformly integrable. From ...
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0answers
55 views

independence of random variables conditioned on function of those variables

Given: $X_1,...,X_N$ iid $\mathcal{X}$-valued RVs, a map $f:\mathcal{X}^N \to \mathcal{Y}$ and a map $g:\mathcal{X}\times\mathcal{Y} \to \mathcal{Z}$. Does the following make sense and hold true? Is ...
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0answers
44 views

Hoeffding's lemma for vector valued RV

I have a random variable $Z$ taking values in $\mathbb{R}^d$, $E[Z] = 0$ and $||Z|| \in [a,b]$ Can I apply Hoeffding's lemma using the random variable $X = ||Z|| - E[||Z||]$ such that $$ \mathbf{E} ...
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1answer
66 views

On Integrating the joint probability density function of two random variables

Suppose that the joint probability density function of two random variables $x$ and $y$ is given as $p(x,y)$. We know that the probability density function of $x$ can be found by integrating out $y$ ...
2
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1answer
42 views

Do paths of a continuous time martingale always have a left limit?

I'm looking at this theorem: Let $(X_t, \mathcal{F}_t)_{t \in \mathbb{R}_+}$ be a submartingale, where $\mathcal{F}$ is right-continuous and complete, and the function $\mu(t) := E[X_t]$ is ...
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0answers
66 views

Rate of convergence of Bayesian posterior

Suppose a data generating process (DGP) is parameterized by some unknown parameter $\theta_0$, say $P_{\theta_0}$, and we want to estimate the value of $\theta_0$ using Bayesian method. Let ...
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1answer
38 views

How can I prove that $E[(Y-E[Y|X])^2]=E[Y^2]-E[(E[Y|X])^2]$?

Given $(X,Y)$ random vector $\Bbb R^2$-valuated, absolutely continous, with known density $f_{X,Y}$ (but I don't think we really need this) I have to exploit conditional expectation properties to ...
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1answer
36 views

Mutual Information Entropy Inequality

I am trying to prove $H(x,y:z)>H(x:z)+H(y:z)$ and here is what I have. LHS: $=H(xy)-H(xy|z)=-\Sigma p(xy)lg(p(xy))+\Sigma p(xy|z)lg( \frac{p(xyz)}{p(z)})$ RHS: ...
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34 views

Identifying a Markov chain

This is a very basic question in the theory of Markov chains and I'm just not sure how to prove it mathematically. Say we have random variables $X, Y$ that are correlated and we have a possibly ...
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0answers
56 views

Ito's formula applied to a stochastic function

The Ito's formula stated in my book is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and $(X_t)_{t \geq0}$ is a ...
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1answer
210 views

Are there any examples of a random variable with infinite second moment and finite variance?

The theorem in my notes says that: If $|X - a|^\alpha$ has expectation for any $a \in \mathbb{R}$, and any $\alpha > 0$, then $|X - b|^\beta$ has expectation for any $b \in \mathbb{R}$ and any ...
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0answers
49 views

Tightness of probability measures

Prove: If there is a $\phi(X)\geq0$ such that $\phi(x)\rightarrow \infty$ for $|x|\rightarrow \infty$ and $\sup_n\int\phi(x)dF_n(x)<\infty$ Then $F_n$ is tight. The definition of tightness of ...
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1answer
49 views

A quick question on Conditional Expectation

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ be a sub-$\sigma$-field. Also let $X$ be an integrable random variable, and $E(X|\mathcal{G})$ the conditional ...
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1answer
20 views

Probability theory - Combinatorics combined with probability

My question concerns the b) part of this one. the answer from a) is needed and i've provided it below. At a group assignment in a highschool, 8 students are divided into two equally large groups. a) ...
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1answer
24 views

Probability theory - Combinatorics

to avoid inmates becoming close friends, they are put into groups where they end up with other inmates. In how many ways can the prison warden divide 6 inmates into a) two equally large groups? b) ...
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1answer
32 views

How to show that the random variables given are independent?

Prove that a sum of random variables say $\sum_{i=1}^{\infty }a_{i}X_{i}Y_{i} $ is independent of the sequence $\left\{ Y_{i}\right\} _{i=1}^{\infty }$ where both the random variables $\left\{ ...
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1answer
23 views

Three-Perpendicular Theorem for linear regressions

For a random vector $X=(X_1,\ldots,X_p)'$, we define $$ \mathcal{L}(X)=\{b_0+b_1X_1+\cdots+b_pX_p,b_0,\ldots,b_p\in\mathbb{R}\}. $$ The linear regression of the $q$-dimensional random vector ...
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3answers
168 views

Need Critique on my solution. Convergence in probability of a product of sequences of random variables

Came across this problem in my self education. Found 4 solution here, but none looked simple enough to me... So I cooked up one of my own. Now it looks too simple :) Am I missing something? I would ...
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1answer
24 views

The relationship between CDFs of two random variables

Suppose that for random variable $X$, we have $Pr(X>b)>p$ where b and p are real numbers and $0 <= p<= 1$. Now suppose that for random variable Y, we have $Y>aX+cX^2$. Can we conclude ...
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2answers
198 views

A problem about the limit of measure

Problem: If $\mu$ is a $\sigma$-finite measure on $(R,\mathscr{B}(R))$, then define $\mathscr{A}$ to be the collection of all $A\in\mathscr{B}(R)$ such that the following limit exists and is ...