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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29 views

Conditional Expectation under Three Different Measures

I'd like to check my understanding on the following, and I'd appreciate a thorough combing with the rigor comb. I'm following the notation in the excellent answer given here. In the end, I'd like to ...
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1answer
32 views

what would be power series of $x_t = e^{\beta_t} $ if $\beta_t$ is a Brownian motion process?

In general the power series of $e^x =1+x/1!+x^2/2!+x^3/3!+...$ but because the process is random we can't apply the direct differentiation than how can i write it's power series.In the book stochastic ...
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17 views

Continuous dependence on an initial condition (SDE)

Let's say I have a (one-dimensional) diffusion process $$dX=\mu(X_t)dt+\sigma(X_t)dW.$$ Assume we have fixed $\epsilon > 0$ and $t >0$ Under what conditions is $\mathbb{P}^x(X_t < ...
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2answers
61 views

Proof: $Y$ stochastically dominates $X$ implies $E[\phi(Y)]\geq E[\phi(X)]$ for increasing $\phi$

Suppose $X$ and $Y$ are real random variables with CDF $F$ and $G$ such that $F(x)\geq G(x)$ (i.e. $Y$ exhibits (first-order) stochastic dominance over $X$). Then, for all increasing function ...
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31 views

Absolutely Continuous Weakly Convergent Sequence Need Not Converge Strongly

The following appears as an exercise in Sinai and Koralov's Theory of Probability and Random Processes. Give an example of a family of probability measures $P_{n}$ on ...
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1answer
31 views

On a step of a proof of the Borel-Cantelli lemma.

This is an excerpt taken form Probability with Martingales by Williams. The framework is probability theory. Why is the equation being discussed true if condition $\{ n \ge m \}$ is replaced by ...
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22 views

Name for a constrained Poisson-like bridge process

I have a sequence $t_i$ for $i=0,2,\cdots,n$ of integer jump times with $t_0=0$ and $t_n=n$ such that the waiting time $t_{i+1}-t_i$ has distribution density $f_i(t)$. So it's kind of like a Poisson ...
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42 views

What is meant by $P(X = x, Y = y)$?

Let $X$ and $Y$ be two random variables. Then $$ P_X(X = x) = P_X(X^{-1}(x)) $$ and $$ P_Y(Y = y) = P_Y(Y^{-1}(y)). $$ But what is meant by $$ P(X = x, Y = y)? $$ I see this notation sometimes ...
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1answer
37 views

Why Does $\rm{E}[1_A \mid X] = P^X(A\mid X)$?

Following the excellent answer here, it is stated that Connection: Let $P^X(\cdot\mid\cdot)$ be a regular conditional probability of $P$ given $X$. Then for any $A \in \mathcal{F}$ we have $$ ...
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1answer
60 views

Dominated convergence theorem for complex-valued functions?

Suppose there is a sequence $\{f_n(x)\}$ such that $\lim_{n\rightarrow\infty}f_n(x)=f(x)$. I've previously used the dominated convergence theorem for interchanging the limit and the integral in ...
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1answer
23 views

$k$ points of contact for percolation

In Grimmett's book on percolation near the beginning of Ch. 7, he summarizes the plan of proof of the result on percolation of slabs. We are in $\mathbb{Z}^d, d\ge2$, with his usual notation that ...
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1answer
24 views

Using the dominated convergence theorem to bound the integral of a random variable

The following claim is used in the solution to problem 9.4 in Jacod and Protter's Probability Essentials: Claim: Let $X\in\mathcal{L}^{1}$ on $(\Omega,\mathcal{A},P)$ (where $\mathcal{A}$ is a ...
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0answers
54 views

A Simple Inquality with Expected Value

Given $X(k) \ge -1, \;\;k=1,2,3...,N$ be discrete random variable with distribution $f_X$ and assume $a \in [0,1]$ so that $aX(k) \ge -1$ for all $k$ and $$ 1 - E \left[ \prod_{i=1}^N(1+aX(i)) ...
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30 views

Conditional expectation and conditional variance decomposition

I am preparing an exam and I have found in my lecture notes the two following formulas that the professor uses again and again, but I have no clue where they come from: ...
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1answer
50 views

Show that $X_n = n \min(T_1,T_2 \cdots T_n )$ has assymtoticaly an exponential distribution as $n \rightarrow \infty$

Let, $T_1,T_2 \cdots T_n $ be i.i.d random variables having reliability function: $R-(t) = 1 - \lambda t - o(t)$ as $t \rightarrow 0$. Show that $X_n = n \min(T_1,T_2 \cdots T_n )$ has ...
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1answer
89 views

Expectation of an increasing transformation of a random variable

Suppose $X$ is real-valued random variable and $\phi$ an increasing function. An upper set is either an open or a closed right half line. Below, all expectations are assumed to exist and $I$ denotes ...
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35 views

Probability intersection of events

Let the sample space be S = 1,2,3,4,5,6 with probability function p. and $p(1)=p(2)=p(3)=p(5) = \frac{1}{8}$ and $p(4)=p(6) = \frac{1}{4}$ Define the events $E = 2,4,6$ and $F = 2,3,5$ Give the ...
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1answer
22 views

Vanishing measure sets and Expectation

During my research, I was required to prove a particular result. I shall just ask what I needed for my result to hold. Let $X_n$ be a sequence of random variables that are integrable and suppose we ...
3
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1answer
147 views

CLT for independent, but non-identically distributed exponential variables

This problem is practice for my qualifying exam and comes from Resnick, chapter 9. Could anyone comment on my solution(s)? Problem Suppose ${e_n, n\ge 1}$ are independent exponentially distributed ...
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3answers
71 views

Complete convergence not happening but convergence in probability occurs

So today I created a counterexample to "Convergence in Probability implies Almost Sure Convergence". I considered a sequence $\{X_n\}$ of independent random variables defined by: ...
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1answer
86 views

On the central limit theorem

The Central Limit Theorem states for a sequence of i.i.d. random variables $\{X_i\}$, $$\frac{\overline{X} - \mu}{\sigma/\sqrt{n}} \to N(0,1)$$ in distribution as $n \to \infty$. I saw in some ...
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1answer
35 views

Is every continuous CDF the limiting distribution of some sequence of discrete CDFs?

Note: I know that for various measure-theoretic reasons (that I don't fully understand) this does NOT apply to the underlying probability density. I'll accept as answers either a proof, paper to a ...
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2answers
41 views

Do two almost surely equal random variables necessarily have the same probability?

Let $\Omega$ be a probability space with $\sigma$-algebra $\mathcal{A}$, and let $\mathcal{B}$ be the Borel $\sigma$-algebra. Let $X:(\Omega,\mathcal{A}) \to (\mathbf{R},\mathcal{B})$ and ...
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1answer
28 views

Mean Squared Displacement of Biased Random Walk [closed]

If $x_t=x_{t-1}+\mathcal{N}(\mu,\sigma)$ and $x_0=0$ what's the value of $\langle x_t^2\rangle$?
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1answer
41 views

Is conditional entropy ever taken to be a random variable?

In probability theory, the conditional expectation $E(X|Y)$ and variance $V(X|Y)$ er usually taken to be random variables, st. the value of $E(X|Y)$ depends on what value $Y$ ends up taking. I've ...
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0answers
51 views

Example tripped Kolmogorov and Wiener

Assuming the hint is true, I attempt to prove the latter prop: Assume on the contrary that $\mathscr{L} = \mathscr{R}$. If $\sigma(Y_0) \subseteq \mathscr{L}$, then $\sigma(Y_0) \subseteq ...
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1answer
43 views

Name when two functions are equal under integration (expectation)?

What is it called when $E[X] = E[Y]$? That is, $$\int x f(x)\,dx = \int y g(y)\,dy.$$ What I want to say is not that the expectation of $X$ is equal to that of $Y$ but rather (the equivalent ...
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1answer
39 views

Definition of independence of infinite random variables

When random variables $Y_1, Y_2, ... Y_n$ are independent, we say that $$P\left(\bigcap_{i=1}^{n} (Y_i \in B_i)\right) = \prod_{i=1}^{n} P(Y_i \in B_i)\tag{F1}$$ or for any distinct indices $i_1, ...
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29 views

Where to find “non-standard” characteristic functions?

Well, the title says it all. I need the characteristic function of the (generalized) arcsine distribution. I desperately searched the internet for it but haven't found anything. Is there some standard ...
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1answer
29 views

What is the distribution of the sum of several normally distributed random variables?

Let's say we have n normally distributed random variables all with the same median and variance. Do we have a possibility to estimate the distribution law of the sum of those variables? I assume ...
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0answers
32 views

Infinitely Many Bernoulli Sums

Suppose $Y = X_1 + X_2 ...$ where $X_n$ is Bernoulli with $p_n$ and that $\sum_{i=1}^{\infty}p_n < \infty$. What is the expectation of $Y$? Since the expectation of $X_n = p_n$ and the ...
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1answer
38 views

Does aperiodicity needed for ergodic theorem to hold true?

Given a irreducible and positive recurrent countable state Markov Chain with unique stationary distribution $\pi$, is the following true $$\frac{1}{n}\sum_{i=1}^{n}X_i \to \sum j \pi_j$$ Or, to ...
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2answers
43 views

Urn with increasing number of distinct balls

Suppose we have an urn that initially has only one labelled ball inside. At each time step, we flip a biased coin with probability $p$ $(\in(0,1))$ of landing on heads and probability $1-p$ of landing ...
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1answer
37 views

Brownian Motion is almost surely unbounded, and a proof for the discrete Random Walk

If $B_t$ is a Brownian Motion, why we have $$ P(\liminf_{t\to\infty} B_t = -\infty) = 1 $$ and $P(\limsup_{t\to\infty} B_t = +\infty) = 1$? I guess for the following "discrete version" of ...
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2answers
28 views

Continuous marginal distributions do not imply continuous joint distribution

I already proved the other implication. I need to find an explicit example that shows that if there is some random vector $(X,Y)$ and $X$ and $Y$ have both continuous marginal distributions, then ...
3
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1answer
46 views

Expected number of return for a random walk on a graph

Let $G$ be a simple, connected undirected graph of order $n$ and vertex set $\{v_1,\ldots,v_n\}$ and let $P = (p_{i,j})$ be a $n \times n$ matrix where $$p_{i,j} = \left\{ \begin{array}{ll} ...
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1answer
22 views

Simple Set Operation with Random Variable

Consider $X(\omega) \ge -1$ be a discrete random variable and define an event $$ \{\omega: 1+a X(\omega) \le \varepsilon\} $$ where $a \in [0,1]$ and $\varepsilon \in [0,1]$. I was wondering ...
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1answer
23 views

Convergence in probability of a composite function.

Question: Let $f$ be a continuous function on $\mathbb{R}.$ If $X_n \to X$ in probability, then $f(X_n) \to f(X)$ in probability. The result is false if $f$ is merely Borel measurable. [Hint: ...
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1answer
42 views

prior probability vs a priori probability

What is the difference between "Prior probability" and "a priori probability" Wikipedia have two distinct pages for them. As of my inference i thought "Prior" and "a priori" are same, i.e., P(y) in ...
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50 views

equivalent form of almost sure convergence

Consider random variables $X_1, X_2, \dots$ and $X$ on $(\Omega, \mathcal F, \mathbb P)$. We say that $X_n$ converges to $X$ almost surely if $$\mathbb P\left(\lim_{n \to \infty} X_n =X\right)=1.$$ It ...
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1answer
50 views

On proving that a infinite intersection of truth sets is empty and on the usefulness of almost surely.

I am trying to solve the exercise at the end of this page, the framework is that of measure theory where we are tossing a coin infinitely often so we are working with a probability triple $( \Omega, ...
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87 views

Conditional probability as a primitive concept

Most of the popular axiomatizations/theories of probability define conditional probability as a ratio involving unconditional probability. Therefore, conditional probability is second class to ...
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1answer
54 views

Uniform continuous distribution for cycles.

Let there be $n$ people standing in a circle and holding hands with probability $p$. What is the expectation value $E(X)$ for the number of 'chains' when $p=.5$? For what $p$ is $E(X)$ largest? ...
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23 views

Empirical Quantilfunction as Integral Bound

This is my first post, so please be nice ;) I'll try to outline my problem correctly and whilst keep it as short as possible! I have to deal (for my master thesis) with the integral ...
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2answers
27 views

Variance and Covariance of Linear Transformations

Suppose $X$ and $Y$ are random variables with $E(X)=2, E(Y)=3 Var(X)= 4, Var(Y)=10$ and $Cov(X,Y)=-5$ Find $Var (5X+2Y)$ From my book I know $$Var(5X+2Y)= Var(5X)+Var(2Y)+2Cov(5X,2Y)$$ but after ...
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1answer
33 views

Introduction to Lebesgue Integration for Statistical Use

I am studying statistics at the graduate level and have a moderate background in real analysis however I unfortunately have no experience with Lebesgue integration. Does anyone have some recommended ...
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1answer
26 views

Build a Markov process from a transition semigroup

Suppose that we have a Markov process $X$ with time-homogeneus transition probability function given by $p(t,x,A)$, with $t>0, x\in E, A \in \mathcal E$, where $(E,\mathcal E)$ is the state space. ...
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1answer
35 views

Kullback - Leibler Divergence and the Triangle Inequality

The KL Divergence, or relative entropy for two probability distributions $p,q$ on $\Omega$ is defined as: $$ H(p|q) = \int_{\Omega} p(\omega) \log \frac{p(\omega)}{q(\omega)} d\omega $$ This is a ...
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28 views

Characteristic function of $a/\sqrt(X+b)$ given characteristic function of $X$

Given that one knows the characteristic function of a rv $X$ how can we write the characteristic function of a function of $X$ when that function is $$\frac{1}{\sqrt{X+b}},$$ for $b$ constant? So, if ...
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20 views

Finding the distribution of a dependent function given the mean and coefficient of variation (COV) of the independent parameters.

It is given that $A$ is log-normally distributed with a mean of 0.19 and coefficient of variation of 0.10, and $B$ is log-normally distributed with a mean 0.040 and COV equal to 0.10. Assume that $A$ ...