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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10 views

Density Function of Random Variable Related to Brownian Motion

Above is my question. I've done the first two parts, that's no problem. I'm stuck on finding the density of the rv $R = W_1 / M$. I have got as far as $$g(x,y) = \frac{\partial^2}{\partial x ...
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1answer
35 views

Probability of getting extra heads

Suppose you are tossing a coin $10$ times. You observe that the first $5$ tosses result in all heads. Then what is the probability that you would get $3$ heads in the remaining $5$ tosses? I have no ...
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2answers
40 views

Expectation of $\mathbb{E}(X^{k+1})$

I have difficulties with an old exam problem : Let $X$ be a positive random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$. Show that $$\int_0^\infty t^k ...
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1answer
7 views

Countable infinite support of probabilistic measure

Let $E\subset\Omega$ be a countable infinite set. I want to define probability measure $p$ on $\Omega$, such that $p(x)>0 \iff x\in E$ and all $x\in E$ have the same probability. Is that possible? ...
2
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0answers
9 views

Consequence of random walk with positive speed on a graph

Consider a random walk $X(n)$ on a vertex-transitive graph where the random walk has positive speed, i.e., $$ \lim\limits_{n \rightarrow \infty} \frac{d(X(n), X(0))}{n}= \alpha>0$$ almost surely. ...
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0answers
19 views

Hitting time is a stopping time

Can somebody help me proving that the following hitting time is a stopping time? Let $\{X_t\}_{t\ge 0}$ be a real-valued, right-continuous process, adapted to a filtration $\mathfrak{F}$ which ...
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0answers
8 views

Markov chain convergence without total variation norm

Can this theorem be reformulated so that the total variation norm $\|\cdot\|_{TV}$ is not used?
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1answer
16 views

Solving a series in the proof of the expectation of the binomial distribution

I am studying the expectations and variances of the most common distributions. For the binomial distribution the mean is equal to $np$. Considering $p$ and $q$ independent variables and ...
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22 views

Are random variables in a tail $\sigma$-algebra in the same probability space?

When defining a tail $\sigma$-algebra, are the random variables necessarily in the same probability space? I mean, it is meaningless to have one random variable in a different sample space? Need the ...
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0answers
10 views

Is Gaussian $(X_1, X_2)$ optimal for $h(a_1X_1+ a_2X_2+Z_1) - \mu h( b_1X_1+b_2X_2+ Z_2)$?

Let \begin{align} W &= h(X_1+Z_1) - \mu h( X_2+ Z_2) \quad (1) \end{align} where $h(\cdot)$ is the differential entropy function, $\mu\ge 1 $ is a scalar, and $Z_1$ and $Z_2$ are independent ...
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1answer
41 views

Understanding a proof of the strong Markov property of Lévy processes

I don't understand the the last sentence of a proof of the Markov property for Lévy processes given in Jochen Wengenroth's textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008). I will appreciate ...
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1answer
16 views

Conditional probability with max(X, Y)

Let $Y_n=$ the outcome of the $n$-th die roll, let $X_{n+1} = \max \{X_n, Y_{n+1}\}$ with $X_1=Y_1$. What is $P(X_{n+1}=j \ | X_1=i_1, ..., X_n=i)$? I know that it is $P(\max \{X_n, Y_{n+1} \}=j \ | ...
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1answer
31 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
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1answer
26 views

Bayes theorem - is it applicable in any case?

I'm studying the Bayes' Theorem and I have a doubt. In this wikipedia page there's an example of application for the following events: ...
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0answers
9 views

Techniques to prove FDD convergence

When examining a sequence of stochastic processes $(\textbf{X}_n)$, $n\geq1$ convergence of marginals, i.e. $\mathbf{X}_n(t)\to\mathbf{X}(t)$ (in distribution) is often not too hard to establish for ...
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1answer
41 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
2
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0answers
28 views

Hoeffding’s inequality extension

In Hoeffding’s inequality we assume that the random variables $X_i$ ,$i=1,..,n$ are i.i.d. and bounded . Is there any extension to Hoeffding’s inequality for the case that $X_i$ are identically ...
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0answers
13 views

bias reduction when the bias depends on the true parameter

Let's say we estimate a parameter, $\theta$, by $\hat{\theta}$. For this estimator we have the following property that $$\hat{\theta}\to_{p}\theta+f(\theta)$$ where $\to_{p}$ denotes convergence in ...
5
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1answer
49 views

A number-theoretic random walk on the integers

Suppose a random walker starts at $S_0 = 2$, and walks according to the following transition probabilities: If the walker is on the $n$th prime number $p_n$, she moves to either $p_n + 1$ or ...
2
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0answers
25 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
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0answers
30 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...
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2answers
34 views

Eigenvalues of a transition probability matrix

I have read that, for $$I - \alpha P$$ where $I$ is the $n\times n$ identity matrix, $\alpha \in (0,1]$, and $P$ is the transition probability matrix with dimensions $n \times n$, $I - \alpha P$ is an ...
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1answer
38 views

Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
2
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1answer
24 views

Total variation distance is complete

For a given measurable space $X$, $\mathcal{P}(X)$ denotes the space of all the probability measures on $X$. The total variation distance $\rho$ on $\mathcal{P}(X)$ is defined by: for $\mu, \nu \in ...
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1answer
48 views

Question about Measure Theory [on hold]

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...
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1answer
35 views

interpreting wording of probability question

Two dice are rolled, and the sum of the face values is six. What is the probability that at least one ofnthe dice came up a three? I want to make sure that I am interpreting the language right when ...
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0answers
57 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or more? [on hold]

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
1
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1answer
27 views

Notation: the $\sigma$-algebra $\mathcal{F}_\tau^+$

I'm reading a probability textbook on stochastic processes (Jochen Wengenroth's "Wahrscheinlichkeitstheorie", de Gruyter 2008) and the following notation: "$\mathcal{F}_\tau^+$" came up in the ...
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1answer
35 views

Pareto distribution

So I'm given a Pareto distribution with parameters $\alpha >0$ and $k>0$ which is the form of \begin{equation*} f(x) = \frac{\alpha k^\alpha}{x^{\alpha+1}},~x > k. \end{equation*} I found ...
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1answer
37 views

Measurability of a set

This question is from Karatzas's Brownian Motion and Stochastic Calculus page 108. Let $W_t$ be standard one-dimensional Brownian motion then it concludes that the set ...
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0answers
28 views

Branching Process in simple random walk

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ ...
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0answers
23 views

Question concerning invariant distribution

Let us consider the Markov chain $(X_n)_{n \in \mathbb{N}}$ with state space $I = \{0,1\}^m$ and transition probabilities $$ p_{xy} = \begin{cases} m^{-1} &\mbox{if } \vert x - y \vert = 1 \\ 0 ...
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0answers
10 views

probability generating function moments for the multivariate case

Suppose ${\bf X} = (X_1, \ldots, X_d)$ is a non-negative integer-valued random vector, with pmf $p$, I tried to extend the results of the univariate generating function to the multivariate case, is ...
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2answers
40 views

proof of conditional probabilities

show that if the conditional probabilities exist then $$p(A_1\cap A_2 \cap \cdots \cap A_n) = p(A_1)p(A_2\mid A_1)p(A_3\mid A_1\cap A_2)\cdots p(A_n\mid A_1\cap A_2 \cap A_3\cap\cdots\cap A_{n-1})$$ ...
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1answer
29 views

probability of 26 letters

A monkey at a typewriter types each if the 26 letters of the alphabet exactly once, the order being random. A. What is the probanility that the word HAMLET appears somewhere in the string if letters? ...
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1answer
53 views

A measure theory question-1 [on hold]

Let $ (\Omega, \mathcal U, P)$ be a measure space and any events $A_1, A_2, A_3 \in \mathcal{U}$ And $ B$ is defined as event of occurrence of at least one of these three events. First I need to ...
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2answers
43 views

A probability theory question [on hold]

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
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3answers
36 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
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2answers
56 views

A question related to measura space

Let a real value $X$ be a random variable and consider $\int_{\Omega}|X|dP \lt \infty $. I need to show that \begin{equation*} nP(|X|\gt n)\to_{n\to \infty} 0. \end{equation*} please help me ...
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0answers
30 views

$\overline{X} \rightarrow c$ in probability $\Rightarrow g(\overline{X}) \rightarrow g(c)$ in probability

Prove that if $\overline{X} \rightarrow c$ in probability and if g is a continuous function, then $g(\overline{X}) \rightarrow g(c)$ in probability. Once I think in the situation $|g|<M$, it is ...
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0answers
22 views

Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
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2answers
21 views

Can't find intersection of two probabilities.

I have the following problem: While producing goods, defect through event A has 3% probability and defect through event B has 4% probability. Total goods that are not defected - 95%. Find correlation ...
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0answers
26 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
2
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1answer
30 views

Intuition/proof that $E(X)= \int X(w) dP = \int x d\alpha$, where $\alpha$ is the cumulative distribution function of X

Looking for more intuition/help explaining the equivalence of the following two integrals. I know that the push-forward measure, or the CDF, of a random variable $X$ on a prob. space $(\Omega, \cal ...
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2answers
28 views

Determing a transition probability matrix

I need some support with this homework exercise: An urn contains at most $N$ balls. Let $X_n$ be the number of balls in the urn after the $n$-th execution of the following procedure: If the urn is not ...
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0answers
22 views

independence and characteristic functions [duplicate]

Why is it that \begin{equation*} \mathbf{E} [e^{i t_1 X_1} e^{i t_2 X_2}] =\mathbf{E} [e^{i t_1 X_1}]\mathbf{E} [e^{i t_2 X_2}] \end{equation*} for RVs $X_1, X_2$ and all $t_1, t_2\in\mathbb{R}$ ...
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23 views

First moment inequality and time-average limits

Suppose $\{A(t)\}_{t \geq 0}$ and $\{B(t)\}_{t \geq 0}$ are two non-negative stochastic processes such that $$ \frac{1}{T} \int_{s=0}^T A(s) \, {\rm d} s \stackrel{\text{a.s.}}{\rightarrow} a \in ...
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1answer
21 views

Show intersection of two algebras are not a $\sigma$-algebra

I have the following question: $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra? I believe the answer is no. I ...
0
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1answer
13 views

convergence of continuous mapped RVs

This is an extension of the result in my textbook, I'm wondering if it's true and if there are any references to it's proof. Let $X_n$ be a sequence of random vectors in $\mathbb{R}^d$, let $g : ...
1
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1answer
103 views

Confidence Interval for Pareto Distribution

A random variable is said to have probability density function $$f_X(x)=\frac{\alpha k^\alpha}{x^{\alpha +1}},\quad \alpha , k>0 \; \text{ and }\; x>k.$$ 1. Compute the MLE estimators ...