Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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5 views

Expectation of a Random Variable multiplied by a Conditonal Expectation.

In a probability book I'm reading (Jacod and Protter) it states: Let $Y \in L^2(\Omega,A,P)$, and $\mathscr{G}$ a sub $\sigma$ algebra of $A$. Then the conditional expectation of $Y$ given ...
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0answers
9 views

Expected value and General distribution Function

Given an iid random variable, the expected value is usually defined as follow: $E[T]=\int_0^\infty t f(t) dt $ where $f(t)$ is the pdf of $T$. On a book I found this definition (without any other ...
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7 views

Showing expectation conditioned on a $\sigma$ field is equal to expectation on a $\sigma$ subfield

The problem I'm working on is: If $X \in L^1(\Omega,\mathscr{F},P)$, and $\mathscr{G},\mathscr{H}$ are $\sigma$ sub-algebras of $\mathscr{F}$, with $\mathscr{H}$ independent of $\sigma (\sigma ...
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1answer
3 views

Markov Processes: $P_x$ and $E_x$

In the study of Markov processes, one usually introduces the measures $P_{\pi}$ on the path space of the process where $\pi$ is an initial distribution of the process $X$ i.e $\pi=\mathcal L(X_0)$. ...
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0answers
8 views

Proof all possible unions of a collection of sets is a sigma algebra

If one lets $\Omega$ be a probability space, and S = $\{A_1, A_2, ... \}$ be a collection of subsets of $\Omega$, then I would like to prove that the set of all possible unions of elements in S is a ...
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3 views

Equivalent Formulation of Markov Property for Homogeneous Chains

In Shiryaev's Probability (just above the strong Markov property, p.568), the author says that an equivalent formulation of the usual Markov property for homogeneous chains is $$P[\theta_nX\in B\mid ...
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6 views

Strong Markov Property for Discrete Stopping Times

I'm having a hard time deciphering a particular proof of the following strong Markov property. Theorem (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb ...
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1answer
22 views

Problem with topological space in probability theory. [on hold]

Let $(X, \tau)$ be a topological space. a) Show that arbitrary intersections of closed sets are closed. b) Prove that a set $F \subseteq X$ is closed if and only if for all sequences $\{x_{n}\} ...
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0answers
17 views

Combining large number of independent probabilities

I am trying to calculate likelihood of laser scan($Z$) at give pose($x$) with known map ($m$) using beam based model $P\left(z_t|x_t,m \right)=\prod_{i=1}^{n}P'\left(z_i|x_t,m \right)$ My scan ...
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1answer
15 views

$n$-step transition probability of a Markov chain

Let $(X_t)_{t\in\mathbb{N}_0}$ be a time-homogenous Markov chain over a finite state space $\left\{1,\dots,m\right\}$, so that we've got $$\Pr(X_{t+1}=j\mid X_t=i_t,\dots,X_0=i_0)=\Pr(X_{t+1}=j\mid ...
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2answers
16 views

Comparing Squared Difference of a Random Variable and it's mean, or it's mean conditioned on a $\sigma$-field.

I have $X \in L^2$, and I want to show $E[(X-E[X|G])^2] \leq E[(X-E[X])^2]$ All I've tried is expanding and simplifying, which gives me wanting to show: $E[E[X|G]^2]-2E[XE[X|G]] \leq E[E[X]^2] - ...
2
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1answer
31 views

What is the bound on $E\|Y_n\|^4$ in terms of $n$?

Let $X_n,n\in\mathbb{N}$ be i.i.d. zero-mean random variables in some separable Hilbert space with $E\|X_n\|^8<\infty$ and $Y_n=\frac{1}{n}\sum_{i=1}^nX_n$. I need to find bounds on $E\|Y_n\|^4$. ...
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1answer
15 views

The “on $\left\{ \tau <\infty \right\}$” in the Strong Markov Property

The strong Markov property is often formulated as $$P[\theta _{\tau}X\in A\mid \mathscr F_{\tau}]\overset{\text {a.s on }\left\{ \tau <\infty \right\} }{=}P_{X_\tau}(X\in A)$$ What exactly does ...
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0answers
12 views

Expectation in Hidden markov process [on hold]

Let the hidden state variable be $S_t\in \{0,1\}$, and the observed variable be $X_t$. How can we prove that $\mathbb{E}(P(S_t=1|\sigma(X_{t-1},...,X_0)))= \frac{p_{01}}{p_{01}+p_{10}}$, where ...
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2answers
17 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
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0answers
30 views

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
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0answers
21 views

a.s. and $L^1$ convergence results related to UI martingales [on hold]

Really stuck on this question: For $x \in [0,1)$, we define for non-negative integers $k,n,$ \begin{equation} b_n (x) = k 2^{-n} \quad \text{ if } \quad k 2^{-n} \leq x < (k+1)2^{-n}. ...
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1answer
25 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
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3answers
47 views

Is it even possible to find the variance of this moment generating function?

This is my moment generating function: $M_x(t) = \frac{6e^t}{t^2} + \frac{6}{t^2} + \frac{12e^t}{t} - \frac{12e^t}{t^3} + \frac{12}{t^3}$. I have to find the mean the variance of it. After taking ...
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0answers
14 views

Finding a sufficient statistic for an iid sample of the Gumbel distribution

$G(x;\alpha, \beta) = \exp\{-\beta e^{-\alpha x}\}$ for $x \in \mathbb{R}$ is a distribution (Gumbel family). Side question: is $G(x;\alpha, \beta)$ a member of the exponential family? I do not think ...
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1answer
25 views

An inverse problem on tail probability

This is a question out of curiosity. Assume that $f(x)$ is a density function for which there is a constant $C>0$ so that $$ \int_t^\infty f(x) dx \le C f(t) $$ holds for large enough $t>0$. My ...
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3answers
37 views

stuck on probabilty question [on hold]

Two technicians are discussing the relative merits of two rockets. One rocket has two engines, the other four. The engines used are all identical. To ensure success the engines are somewhat redundant: ...
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1answer
17 views

Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...
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0answers
17 views

Irreducible Markov chain being recurrent

I've come across the following theorem in Sheldon Ross's book whose converse part I am unable to prove. Theorem: An irrreducible Markov chain with state space 0,1,2,... is recurrent if and only $\ ...
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0answers
22 views

How to select a strongly convergent subsequence from a weak convergent sequence in $L^2$?

Let $(p_n)_{n \in \mathbb N}$ be a sequence of probability density functions, which satisfies i)$$ \partial_t p_n (t,x) = \partial_{xx} (a_n (t,x) p_n (t,x)), $$ where $a_n$ is a sequence upper ...
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1answer
36 views

Conditional Expectation and where a RandomVariable is Infinite

My question here is: If $Y$ is positive on $\{\Omega, A, P\}$ and $G$ is a $\sigma$ subfield of $A$ show that $\{Y=+\infty\} \subset \{E[Y|G]=+\infty\}$ a.s. I'm not even sure how to begin solving ...
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2answers
65 views

How can we apply the Borel-Cantelli lemma here?

Let $(A_n)$ be a sequence of independent events with $\mathbb P(A_n)<1$ and $\mathbb P(\cup_{n=1}^\infty A_n)=1$. Show that $\mathbb P(\limsup A_n)=1$. It looks like the problem is practically ...
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1answer
29 views

Filtration right continuity completion

I have a question about filtration. Now fix a measurable space $(\Omega,\mathcal{M})$. Let $(\mathcal{M}_{t})_{t\in[0,\infty)}$ be a filtration on $(\Omega,\mathcal{M})$. We set \begin{eqnarray*} ...
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2answers
33 views

Probability of begin the first of two players to get H when tossing a coin

Two players, $A$ and $B$, alternately and independently flip a coin and the first player to obtain a $H$ wins. Assume player $A$ flips first. Suppose that $P(H)=p$. Show that for all $p$, ...
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3answers
51 views

A and B are independent under$\mathbb{P}$ but not under $\mathbb{Q}$

As the title, how to construct such an example that 2 events from the same measurable space ($\Omega$,$\mathscr{A}$) are independent under probability measure $\mathbb{P}$ but not independent under ...
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1answer
23 views

Is there a shorter way for me to answer this joint probability question?

Suppose a box contains 10 green, 10 red, and 10 black balls. We draw 10 balls from the box by sampling with replacement. Let X be the number of green balls, and Y be the number of black balls in the ...
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2answers
28 views

Random variable $X^2$ determined by moments

Let $X$ be a real random variable, with standard normal distribution. Is the distribution of $X^2$ determined by its moments? In general, if $n \in \mathbb N$, is the distribution of $X^n$ ...
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0answers
38 views

optimization problem gaussian maximizes entropy

Let $X_1, X_2, Z_1$ be random variables and define $$Y=aX_1+bX_2+Z_1$$ I have the following optimization problem of difference of entropies, $$f=\max_{p(x_1x_2)} h(Y) - h(Y|X_2)= \max_{p(x_1,x_2)} ...
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0answers
19 views

Exchangeability and pairwise exchangeability

Suppose we have $\{Y_{i}\}_{i=1}^{n}$ that is exchangeable so the joint distribution of this sequence is the same as $\{Y_{\sigma(i)}\}_{i=1}^{n}$, where $\sigma:\{1,...,n\}\rightarrow\{1,...,n\}.$ ...
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0answers
32 views

Continuity preserved unter expectation? Dominated convergence?

Let $Z$ be a random variable with $0<Z<\infty, 0<\mathbb{E}[Z]<\infty$ and $Z$ be atom-less, i.e. $\mathbb{P}(Z=z)=0$. Further, let $g:\mathbb{R}^+\to\mathbb{R}^+$ be continuous and ...
1
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1answer
13 views

Proof of convergence in distribution

I want to prove that a sequence of random variables $X_n$ converges in distribution to $N(0,1)$, if we have the following condition for an arbitrary $\epsilon>0$: $$(1-\epsilon)Y_n \le X_n \le ...
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0answers
17 views

Simulating beta random variables

Let's say I estimated the parameters of Beta distribution with a single-period dataset, and want to generate a multi-period sample. How can I do that? To be more specific, I estimate my parameters ...
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1answer
27 views

Prove of Stopping time

Let $(X_k)_{k\in\mathbb{N}}$ be iid random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$. Let $Z_n=\prod_{k=1}^n(1+X_k)$, so $Z_n$ a martingale. Consider ...
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1answer
61 views

Probability Question- A Poker Game

Jesse and three of his friends are playing Poker with 32 cards, The 32 cards are of every combination of the for patterns with the numbers 1, 7-13. In this game, each player takes five cards randomly. ...
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1answer
22 views

Derivative of integral over part of Gaussian distribution

I am currently trying to compute the following derivative and integral: $$ P\psi_\theta = \frac{d}{d\theta}\int_{-k}^k tf_T(t)dt, $$ where $t=x-\theta$ and $X\sim N(\theta_0,\sigma^2)$. $f_T$ above ...
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1answer
30 views

The unit circle(disk), $\sigma(X)$ measurable function

If I have two measurable functions $X,Y:S \to \mathbb{R}$ (with the Lebesgue mesure) such that $X(\{x,y\})=x$ and $Y(\{x,y\})=y$ on the unit circle that is $x^2+y^2=1$. Then is $Y$ ...
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0answers
25 views

Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
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1answer
22 views

Variance from the pdf

FOr the interpretation of the variance, it is the fluctuation of the data around the mean. So if I know that mean (say mean=0), and then there are lots of data (70%) points are greater than +/-10 away ...
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0answers
36 views

Partial sums are alternate upper and lower bounds for $\mathbb{P}(\cup A_i)$

Show that $$ \sum_{k=1}^m(-1)^{k+1} S_k \leq \mathbb{P}(\cup_{i=1}^n A_i) \leq \sum_{k=1}^{m'}(-1)^{k+1} S_k$$ where $m, m' \leq n$, $m $ is even and $m'$ is odd, and $S_k = ...
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0answers
31 views

How to prove that the distribution function Fx is left continuous if and only if the distribution law µ is non atomic [on hold]

How to prove that the distribution function $F_x$ is left continuous if and only if the distribution law $\mu$ is non atomic. Can the law $\mu$ and lebesgue measure be singular if the distribution ...
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1answer
29 views

Probability and Induction help [on hold]

Let $Y=X_1+X_2+ \cdots+X_n$ where $X_1, X_2, \ldots, X_n$ are independent Bernoulli random variables, each with probability of success equal to $q$. Use induction to prove that $Y$ has a Binomial ...
2
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1answer
28 views

A linear combination of characteristic functions is a characteristic function?

Let $\phi_k(t)$ be the characteristic function of a random variable $X_k$, $k = 1,2,\dots$. Consider a set of positive real numbers $\{p_1, p_2, \dots \}$, take a function: $$\phi(t) = ...
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2answers
31 views

Let $A$ and $B$ be events, and let $I_A$ and $I_B$ be the associated indicator random variables [on hold]

Let $A$ and $B$ be events, and let $I_A$ and $I_B$ be the associated indicator random variables. Show that : $I_{A\cap B} = I_AI_B = \min(I_A, I_B)$ and $I_{A\cup B} = \max(I_A, I_B)$
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2answers
27 views

Proof Question- Need Help [on hold]

Show that if $P(A|E) \geq P(B|E)$ and $P(A|E^c) \geq P(B|E^c)$, then $P(A) \geq P(B)$. I am reviewing for test, and I came across this problem in the textbook. I need help with this question.
3
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0answers
59 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...