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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How is the binomial distribution connected with the theoretical approach to probability?

I've been told the theoretical approach to probability is defined as follows $$\operatorname{Pr}(\textsf{something})=\frac{\textsf{Favorable events}}{\textsf{possible events}}$$ This has to be ...
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1answer
31 views

Proving a simple aspect of the CDF for discrete functions F(b) - F(a) [closed]

Let $F_X$ be the cumulative distribution function for discrete random variable $X$. How would you prove this: $$ F_X(b) − F_X(a) = \operatorname{Pr}(a \lt X \le b) $$
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1answer
21 views

Are 2 dependent probabilities always disjunct?

If 2 probabilities are disjunct then they are not independent. Are any 2 dependent probabilities also disjunct? Modus ponens for instance are 2 dependent and mutually exclusive events (A and B) where ...
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0answers
25 views

Is there a continuous version of the Borel-Cantelli lemma?

Given a sequence of events $A_n$ for $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limes ...
2
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1answer
21 views

About modifications of right-continuous stochastic processes

Lemma : Let $X$ and $X'$ be two right continuous(or left continuous) processes defined on the same probability space $(\Omega,F,P)$ be a modification then the two processes are indistinguishable. ...
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24 views

Convolution of two random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
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1answer
30 views

conditional distribution of X, given Y=y and Z=z, and compute E(X|y,z)

Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere. Find the conditional distribution of $X$, given $Y=y$ and $Z=z$, and compute $E(X|y,z)= ...
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1answer
24 views

Portmanteaus Theorem and Lipschitz functions

I was looking at some basics in the theory of convergence of distributions, e.g. Portmanteaus Thm. One part of it asserts that it suffices to check $\mathbb{E}[f(X_n)]\to\mathbb{E}[f(X)]$ for bounded ...
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1answer
37 views

Probability of $P(X_1 X_2\le 2)$

What is the probability of $P(X_1 X_2\le 2)$. Both variables are independent and each has the probability density function $f(x)=1, 1<x<2$, zero elsewhere. First I would like to assume that the ...
2
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1answer
28 views

Expectation of proportion random variable

Suppose that $X$ and $Y$ are two random variable. $X$ and $Y$ are independent. We also have $f$ and $g$ are two continuous functions on $\mathbb{R}$. Is the following equation true? $$ ...
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0answers
10 views

Calculate Expectation of points in a homogenous poission process with parameter $\alpha $ as a renewal process?

If a poisson process $N $ on $[0, \infty ) $ has rate $\alpha $ (ie $E N(A)=\alpha m(A) $, $m $ lebesgue measure ) can its points be represented as occurences in a renewal process with interarrival ...
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0answers
45 views

How to prove that $X+Y \mod p$ is indpendent from $X$ if $X$ and $Y$ are independent?

We have a group $\mathbb{Z}_p$ and some random variable $X$ and $Y$ with this domain. We have that $Y$ is chosen uniformly at random, thus each element from $\mathbb{Z}_p$ with probability ...
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18 views

Deviation From Expectation Question [closed]

Suppose that X1,......,Xn are independent and identically distributed with Xj~X. Show that: Variance(X1+...+Xn) = nVar(X) and Var(nX = (n^2)*Var(X). Conclude that, for n > 1, x1+...+xn is not(~) ...
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1answer
24 views

Finding a CDF from the PDF of another Random Variable

Given a function $$ Y=ae^x $$ With the distribution: $$ f_X(x)=be^{-bx} \,\,\,\, x\geq0 $$ Show that the cumulative distribution function is: $$ F_Y(y)=1-(y/a)^{-b} \,\,\, y\geq c $$ My approach ...
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1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
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1answer
16 views

Using Monotone Convergence Theorem to extend a result involving random variable

We assume that for a non-negative, bounded, continuous random variable we have $$ E[X]=\int_0^\infty P(X>x) dx $$ Now the task is to extend this result to non-negative, continuous random variables ...
2
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1answer
39 views

Alpha mixing property of a $\mathbb{R}^d$ valued Stochastic Process

In statistics and probability literature, a strictly stationary stochastic process $\{X_t\}\in\mathbb{R}$ is called $\alpha$-mixing if $\alpha(n)=\sup_{A\in\mathcal{F}_{-\infty}^{0}, ...
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2answers
72 views

Best advanced book in probability theory?

I have read real and complex analysis, and Probability theory by Feller(Vol1 & 2). What is the next book I should read to go deeper into subtopics of probability theory? Thank you!
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1answer
10 views

Additivity of the Chi-squared distribution

Can additivity of the $\chi^2$-distribution be used when substracting variables? For example: Does X-Y=Z where X~$\chi^2$(3), Y~$\chi^2$(1), Z~$\chi^2$(2)?
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30 views

Creating a random network (graph) with a $\textbf{random}$ number of vertices and given degree distribution

I was trying to find an answer to my question on google scholar, however I didn't find anything that is close to what I am looking for. I would be very grateful for your help. There is a theory of ...
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0answers
39 views

Lindeberg condition of order $2k$

Prove that if $ (X_n),\: n \ge 1$ are independent r.v.s with $ E X_n = 0, E X_n^2 = \sigma_n^2$ which obey central limit theorem and $$\lim_{n\rightarrow\infty}E\bigg(\frac{S_n ...
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0answers
28 views

Property of uniformly tight random variables?

I'm stumped on the following question, which is problem 1.3.9 in the book Weak Convergence and Empirical Proceses by van der Vaart and Wellner. It is based on the following notion of asymptotic ...
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1answer
15 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...
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1answer
34 views

MLE of $n(\theta,a\theta)$ family

Question: A special case of a normal family is one in which the mean and variance are related, the $n(\theta,a\theta)$ family. If we are interested in testing this relationship, regardless of the ...
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0answers
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Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
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3answers
88 views

Doubt about Probability of arranging identical balls

There are four boxes and 12 balls. The boxes are numbered and hence distinguishable but the balls are identical. What is the probability that a random arrangement would result in 10 balls in box 1 2 ...
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0answers
22 views

Can I use the triangle inequality to ensure a unique measure?

Let $\mathcal{C}$ be a finite set of objects, $\Delta\mathcal{C}$ the set of probability measures on $\mathcal{C}$, and $\mathscr{U}$ be a finite set of linear functions, $u: ...
2
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1answer
38 views

Conditional expectation of a random vector taking values in convex sets

on a probability space $(\Omega, \mathcal{F},\mathbb{P})$ i have a random vector $X\in L^1_{\mathbb{P}}(\mathbb{R}^d)$ (integrable with values in $\mathbb{R}^n$), such that $\mathcal{P}-a.s.$ $$X\in ...
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2answers
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Should I avoid distribution functions in probability?

I'm reading Erhan Çınlar's book on Probability and Stochastics, and in Chapter 2, he says that distributions are used extensively in elementary probability theory in order to avoid measures. And ...
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1answer
24 views

What is application of gamma distribution on pure math or probability theory?

What is application of gamma distribution on pure math or probability theory? i saw it on several probability textbook as a definition, but it seems to me mathematician couldn't derived it if it is ...
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1answer
21 views

Composition of random variable with its distribution is uniform

I'm trying to solve the following problem (Exercise 11.13) from Probability Essentials by Jacod and Protter: Let $X$ be a random variable (on $\mathbb{R}$) with distribution $F$ that is continuous. ...
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34 views

Does a discrete random variable on a finite sample space always have an expected value?

Let $\Omega=\{A,B,C,D\}$,$\mathcal{F}=\{\emptyset,\{A,B\},\{C,D\},\Omega\}$, $P(\{A,B\})=1/2$, and $P(\{C,D\}) = 1/2$. Now $(\Omega,\mathcal{F},P)$ is a probability space. Let ...
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Convergence of random variables depending on the measure

Suppose the probability spaces $\left([0,1], \mathcal{B}([0,1],\mu_i \right)$ for $i=1,2,3$ , where $$ \mu_1 = \lambda , \ \ \ \ \ \ \ \ \ \ \mu_2 = \delta_1,\ \ \ \ \ \ \ \ \ \ \mu_3 = ...
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0answers
27 views

Why is the covariance a measure of how much two random variables change together?

In descriptive statistics, the empirical covariance of a point cloud $\left\{(x_i,y_i) : 1\le i\le n \right\}$ is defined as ...
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1answer
39 views

Smallest $\sigma$-algebra and $\sigma$-algebra generated by a function

I'm reading through the following theorem: Let $X=\{X_t,t\in T\}$ be a stochastic process. Then $\sigma (X)=\sigma ( \cup_{t\in T} \sigma (X_t))$ From my basic knowledge of measure theory, I ...
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1answer
22 views

If $\{X_n\}$ is a martingale, then $E[X_n-X_{n-1}]=0$

Apparently this should be quite simple, but I have been trying for a while and can't seem to get this. Let $\{X_n\}$ be a martingale, then we have: $$E[X_n-X_{n-1}]=0$$ According to some notes I found ...
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1answer
17 views

Is truncating a discrete probability mass function possible?

I have random variable X, and probability distribution: $P[X = A] = .4$ $P[X = B] = .3$ $P[X = C] = .2$ $P[X = D] = .1$ I want to create a conditional probability with event F. Where F is the ...
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2answers
25 views

Prove that mean independent random variables are uncorrelated

I need to prove $E(X|Y) = E(X)$ implies $E(XY) = E(X)E(Y)$ Is my proof correct? $ E(X)E(Y) = E(X) \int yf_Y(y)dy $ $= \int E(X) yf_Y(y)dy$ .... ($E(X)$ is a constant) $ = \int(\int ...
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1answer
16 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
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1answer
26 views

When will $\lim_{m \to \infty} Y_m = c \Rightarrow \lim_{m \to \infty} E[Y_m] = c$?

I am reading my probability theory textbook. In one example, the author use the strong law of large numbers to show that a random variable $Y_m$ converges to some numbers $c$. Thus, he concludes that ...
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1answer
68 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
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1answer
96 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
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1answer
18 views

Matching moments implies matching densities?

If $X$ and $Y$ are random variables with matching moments (ie: $\mu_X^i = \mu_Y^i (\forall i \in \mathbb{Z}^+)$ then are the density functions of $X$ and $Y$ identical (almost everywhere)? Idea: I'm ...
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1answer
17 views

stochastic matrix and inner product

Can a stochastic matrix be written as $V^{-1} D V $? V is an invertible matrix and D is diagonal. I think so but I can't think of a good proof. Also, the left eigenvectors and right eigenvectors are ...
2
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1answer
33 views

Understanding the measurability of conditional expectations

My question is about the conditional expectation of random variables with respect to a $\sigma$-algebra. I am having trouble getting an intuition behind the definitions among other things. I know ...
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2answers
26 views

Show convergence in distribution by the continuity theorem

So the problem I'm about to solve is to show that: $X \in \Gamma(a,b)$. Show that \begin{equation} \frac{X-E[X]}{\sqrt{Var(X)}} \xrightarrow{d} N(0,1) \end{equation} as $a \rightarrow \infty$, by ...
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1answer
26 views

Calculation of conditional probability

A problem as following: (from Prob, statistics, and random processes for electric engineering, p.264) If I want to find $P(Y\leq y \mid X=+1)$, it can be calculated as following: $P[N+1\leq ...
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1answer
22 views

A question about random walk similar Markov Chain

This is an exercise from Probability and Measure by Billingsley: Let $(X_n)$ be a Markov chain with $\Bbb{Z}$ as its state space and with transition probabilities $(p_{ij})$ such that $$ ...
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0answers
13 views

Understanding $O_p$ [migrated]

One thing I feel like I have never mastered is the concept of $O_p$ convergence and how to use it. I understand the basic idea and what bounded in probability means, but I always have a hard time ...
4
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4answers
706 views

Gambling problem

Question Robert will win $\$1$ with probability $\frac{1}{4}$, win $\$2$ with probability $\frac{1}{4}$, and lose $\$1$ with probability $\frac{1}{2}$ in a bet. Each bet is independent. Determine ...