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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8 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
23 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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1answer
29 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 ...
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0answers
4 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
34 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
18 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 ...
2
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0answers
24 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 ...
2
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1answer
27 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 ...
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0answers
10 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 ...
2
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1answer
20 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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0answers
32 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
49 views

Question about $M/GI/ \infty $ queue

Consider an $M/GI/ \infty $ queue with the following service time distribution: the service time is $1/\mu_i$ with probabbility $p_i$, and $\sum_{i=1}^kp_i=1$ and $\sum_{i=1}^kp_i/\mu_i=1/\mu$. In ...
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1answer
32 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

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
28 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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0answers
56 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 ...
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1answer
104 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
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1answer
43 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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2answers
54 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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2answers
41 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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1answer
52 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
44 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
7
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1answer
75 views
+100

Mean value theorem for random variables (inside an expectation value)

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$E[f(X+Y)]=E[f(X)+E[f^\prime(X+\theta Y)]Y]$$ for real valued random variables $X$ and $Y$ ...
2
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1answer
36 views

Equation with mean of random variables

In a proof I found the following conversion $$E\left[|X|\mathbf{1}_{[a,b]}(Y)\right] = E\left[|X|P(a \le Y \le b)\right]$$ I understand, why $E\left[\mathbf{1}_{[a,b]}(Y)\right] = P(a \le Y \le b)$, ...
2
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1answer
81 views
+50

Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk: A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
1
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1answer
23 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 ...
0
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1answer
34 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 ...
1
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1answer
71 views

Density functions and estimators

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 ...
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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0answers
22 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 ...
2
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0answers
26 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
9 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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0answers
205 views

Pareto distribution,fisher information, confidence interval [on hold]

Having a bit of problem at these questions, greatly appreciated if anyone can solve them. For the notation, k^ is k with a hat on top of it, don't know how to do that on a keyboard. The rest is ...
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2answers
38 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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0answers
60 views
+50

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
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0answers
10 views

Poisson Distribution Optimization Problem

A retailer buys $n$ cookies and has to pay $\zeta_1$ for each. He wants to sell them for a price of $\zeta_2$ (with $0$ < $\zeta_1$ < $\zeta_2$). Let X be a random variable which states, how ...
2
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3answers
26 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
27 views

Conditional probability with students seating

There are 14 students and 9 of them are friends. Students purchased tickets to movie and they got seats in a row of 14 seats. 8 friends got seats next to each other. What is the probability that ...
0
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1answer
21 views

Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
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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 ...
2
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0answers
25 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 ...
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0answers
29 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
21 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
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 ...
0
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1answer
18 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 $t_1, t_2\in\mathbb{R}$ ...
7
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1answer
994 views

Moment generating functions/ Characteristic functions of $X,Y$ factor implies $X,Y$ independent.

This is solely a reference request. I have heard a few versions of the following theorem: If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ ...
1
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0answers
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
29 views

Limit moment generating function

For n a natural number let $X_{n}$ have discrete uniform distribution on interval {1,2...,n} and $Y_{n} =\frac{1}{n} X_{n}$. I need to show that for all t(real number) the $\lim_{n \to \infty} ...
1
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1answer
19 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 ...