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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41 views

Learning probability from Zero level

How do i develop a very good conception in probability theory to do advanced theoretical research in communication theory?Which book do i follow to gain a solid conception about the subject.I have the ...
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69 views

Show that $\frac{1}{n}X_n\to 0$ a.s.

Show that for any sequence $(X_n)_{n\in\mathbb{N}}\in (L_{\mathbb{P}}^2)^{\mathbb{N}}$ of identically distributed random variables it is $\frac{1}{n}X_n\to 0\text{ a.s.}$. The professor ...
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33 views

Events on different experiments

I have two questions for Probability theory and hope getting your help. Question 1: Consider the following two experiments: $T_1$: "Tossing a fair coin". $T_2$: "Tossing a fair six-sided die". ...
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534 views

Problem on continuous probability distribution

Problem:We pick two random numbers, x and y, between 0 and 2. What is the probability that x*y<1 AND y/x<1. I am familiar with continuous probability distributions for one variable, but it ...
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155 views

Probability/Uniform Distribution

A student has the opportunity to take a test at most thrice. The student knows that each time he takes the test, his score is independent random draw from uniform distribution on interval [0,100]. ...
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44 views

A question about choice of signs in expectation

Let $ f_1,...,f_N $ a collection of fucntions, $ \epsilon_1,...,\epsilon_N$ randomized signs ( $\pm1$) with same probability and $ N\in\mathbb{N}$. If $ ...
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87 views

Given E[X] and Var[X], what is P(X=0)?

Let $E[X]=\theta(n^{1.5})$ , $Var[X]=\theta(n^{2.5})$ , $X \ge 0$ What can you say about $P(X=0)$ using Chebyshev's inequality? A: $P(X=0)=O(1/n^2)$ B: $P(X=0)=O(1/n^{1.5})$ C: $P(X=0)=O(1/n)$ ...
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43 views

Proof for expected value of geometric RV without using derivates or other “fancy” methods

Is it even possible? I'm guessing it is, but I get stuck very early on: $E[X]=\sum_{k=1}^\infty kp(1-p)^{k-1}=\sum_{k=1}^\infty k(1-q)(q)^{k-1}=\sum_{k=1}^\infty k(q^{k-1}-q^k)=\sum_{k=0}^\infty ...
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1answer
45 views

Use of convolutions to compute the distribution of the sample mean?

Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute ...
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58 views

Under what conditions do all singletons belong to sigma algebra?

Let's suppose that (A, F) is a measurable space (A underlying set, F the sigma algebra), and that F arises from some topology T. I would like a theorem of the form: All singletons belong to F if and ...
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188 views

How to define a probability distribution over a function space?

What is the mathematically rigorous way of defining a probability distribution over some function space e.g. $L^1[0,1]$? Edit: After reading about the basics of measure theory, I realized that the ...
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41 views

Average over all positive functions on the unit interval whose Lebesgue integral is one

I want to average over all positive functions on the unit interval whose Lebesgue integral is one. Formally, I want to compute the mean of the following probability distribution defined over function ...
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1answer
129 views

Probabilistic Proof That An Absolutely Continuous Function is Differentiable Almost Everywhere

Consider the probability space $([0,1), \mathcal{B}, \lambda)$ where $\mathcal{B}$ is the Borel $\sigma$-algebra and $\lambda$ is the uniform measure. Let $A_{i,n} = [(i-1)2^{-n}, i2^{-n})$ for $i \in ...
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273 views

Upper bound of Euclidean norm on vectors in $\mathbb{R}^n$

Show that for any vectors $v_1,\ldots,v_n \in \{-1,1\}^n \subset \mathbb{R}^n$, there exist $\epsilon_1,\ldots,\epsilon_n \in \{-1,1\}$ such that the Euclidean norm of $v=\sum_{i=1}^n \epsilon_i v_i$ ...
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107 views

Circle rotation (dynamic system)

Here's a passage of my script I do not understand. Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then $$ T_a:=\Omega\to\Omega, z\mapsto ...
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1answer
91 views

Proving increasing function defined as bivariate normal

Suppose $c>0,\sigma>0$ and $\tau>0$ are fixed real constants. Then I'd like to prove that the function $g_c:(-1,1)\mapsto\mathbb{R}$ defined by \begin{equation} ...
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2answers
385 views

Markov's Inequality, only non-negative random variables

I have a question about a Markov's inequality, which states following. Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable on probability space $(\Omega, \mathscr{A}, P)$ and ...
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1answer
83 views

Jensen's inequality for countable probability space

One form of Jensen's inequality for the finite case, tells us that $$ \sum_{x \in X} p(x) \log q(x) \leq \log\sum_{x \in X} p(x) \cdot q(x) $$ For positive p(x), and $\sum_{x \in X} p(x) = 1$, ...
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95 views

Probability of a sample from a random variable with Gaussian distribution

I am studying a paper [1] which states that, as far as I understand, the probability of a single sample $x$ taken from a random variable $X$ with Gaussian distribution equals the Gaussian distribution ...
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264 views

How can this population extinct because of gender inequality?

This population has the properties as follows: (1) It is an isolated population, which means the individuals can only mate with others in this population. (2) It is a monogamy population, which ...
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1answer
19 views

Preservation of positivity under convergence in distribution?

I have the following situation: $\mathbb{P}(X_n\geq 0)=1\quad \forall n\in \mathbb{N}$ and $X_n \overset{\mathcal{D}}{\rightarrow} X$ as $ n\rightarrow \infty$. How do I prove that the positivity ...
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1answer
45 views

range of a function defined by expectation

Given $c>0$ and define a function $f:(0,\infty)\mapsto \mathbb{R}$ by \begin{equation} f(\sigma)=\frac1{\sqrt {2\pi}\sigma}\int_{-\infty}^\infty\max\{-c,\min\{x,c\}\}^2e^{-\frac{x^2}{2\sigma^2}}. ...
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2k views

Are any linear combination of normal random variables, normally distributed?

It is easy to show that if we have n independent normally distributed random variables, then a linear combination fo them ar normally distributed. It is also said that if (x1,x2,..,xn) is ...
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501 views

Infinite Sum of Normals

If we sum a finite number of Normals, $\displaystyle \sum^n_{k=1} X_k \sim N(\sum^n_{k=1}\mu_k,\sum^n_{k=1}\sigma_k^2)$. Is this valid when we do an infinite sum, as long as $\sum^\infty_{k=1}\mu_k ...
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1answer
52 views

Weak Law of Large Numbers - Why Use $Pr()$ Statement With Limits?

In the literature describing Law of Large Numbers, the law is described in terms of $X_1,..,X_n$ random variables which are i.i.d. with each $X_k$ having mean $\mu$ and variance $\sigma^2$. My ...
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1answer
51 views

Limit of function defined by expectation

Given $c,\sigma,\tau$ are positive real constants and define a function $f:(-1,1)\mapsto \mathbb{R}$ by \begin{equation} ...
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1answer
53 views

almost sure convergence in distribution

Consider an i.i.d sequence $(X_i)_{i \geq 1}$ of scalar random variables distributed as $\pi$. It is not difficult to check that for almost every realization $(x_i)_{i \geq 1}$ of this sequence, the ...
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1answer
44 views

Estimating the overall proportion of people who will vote for an individual after sampling two unique groups

Say you have two disjoint groups of people, $A$ and $B$, in an entire population. Out of the overall population, the proportion of people in group $A$ is $q$. We want to determine what the overall ...
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38 views

Find the probability distribution function of $Y_(n)$ = max($Y_1, Y_2, . . . , Y_n$).

Let $Y_1, Y_2, . . . , Y_n$ be independent random variables, each with a beta distribution, with $α = β = 2$. Find a. the probability distribution function of $Y_(n)$ = max($Y_1, Y_2, . . . , Y_n$). ...
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34 views

A basic question on $L_p$ norm

How to prove that if $\mu(\omega) < \infty$ then $L_p$ norm increases to $L_\infty$ norm ?
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1answer
35 views

Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
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204 views

Prove that $Var(X) =\sigma^2E[N]+\mu^2Var(N)$

Given $Y_1,Y_2,Y_3......$ are iid, random variables with mean $\mu$ and variance $\sigma^2$.Suppose that N is an independent random variable taking positive integer values such that $E[N^2]$ finite. ...
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85 views

Prove $P(X\ge \lambda EX)\ge (1-\lambda)^2\frac{(EX)^2}{EX^2}$ if $X$ positive r.v., finite variance

Prove $\displaystyle P(X\ge \lambda EX)\ge (1-\lambda)^2\frac{(EX)^2}{EX^2}$ if $X$ positive r.v., finite variance and $\lambda\in (0,1)$. I wanted to use Chebyshev's inequality but the sign in the ...
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0answers
46 views

Apparent contradiction when manipulating conditional expectations

For a random vector $(T,S,\theta)$ with joint distribution $f(\cdot)$ define the following function (slightly abusing notation for the distributions ): ...
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43 views

Exposition of Erdős and Rényi's 'New law of large numbers'.

Where can I find an exposition of the paper On a new law of large numbers by Erdős and Rényi? I'm reading this paper and it's rather terse, so I'd like some intuition and explanation. I did a Google ...
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177 views

Are marginal densities always greater than the corresponding joint density?

I.e. if $\mathbb P\left(x,y\right)$ is a joint density function, and $\mathbb P\left(y\right)$ is a marginal distribution is it always true that: $\mathbb P\left(x,y\right)\leq \mathbb ...
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1answer
67 views

A basic question regarding the proof of existence of product measure

Suppose that $(X,\mathcal F_1,\mu)$ and $(Y,\mathcal F_2,\nu)$ are measure spaces and suppose that $\mu$ and $\nu$ are finite measures. Define the function $\nu_E : X \to \Bbb R$ by $$\nu_E(x) = \nu ...
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1answer
53 views

proving increasing function

Given $c>0$. Let $f:(0,\infty)\to (0,\infty)$ be a function defined by \begin{equation} f(x)=\frac1{\sqrt{2\pi}x}\int_{-c}^ct^2e^{-\frac{t^2}{2x^2}}dt. \end{equation} I'd like to prove that $f$ is ...
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1answer
47 views

ergodic theorem in the seasonal component analysis of time series

When studying the "seasonal components" part of time series, I once read the following statement. I do not understand what role does the ergodic theorem play here? The decomposition of the process ...
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43 views

Showing an inequality relating two Poisson tail-probabilities

In my research, I've discovered that a property that I am interested in is equivalent to an inequality involving two tail-probabilities of the Poisson distribution. I belive this inequality to be ...
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448 views

Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
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1answer
32 views

If $X\in\mathcal{L}_P^1$ does then the variance exist?

At the moment I am wondering about the following exercise: Let $X\in\mathcal{L}_P^1$, show that $\mathbb{V}(X)=E(X^2)-E(X)^2$. It's clear how to show that, that's not my point. I am ...
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120 views

The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
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1answer
37 views

A monotonicity property for ratios of power means

Let $Z$ be any non-degenerate positive random variable with pdf $% g(z)$. Let $a>0$ and $r\neq 0$ denote arbitrary real numbers. Define the "$r$ -mean" of $Z$ shifted by the constant $a$ as ...
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73 views

Reference for a proof of which 2-increasing functions are joint cdf's

Can somebody give me a reference giving the detailed statement and proof of the fact that the joint cdf's of positive Borel measures $\mu$ on $\mathbb{R}^2$, so $$F(a,b) = \mu(\{(x,y) : x \leq a, y ...
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1answer
24 views

(possibly?) equivalent defintiioons of indepdnt events

Let $A_1$ and $A_2$ denote two events. Now I'm sure from school we've read that $A_1$ and $A_2$ are called independent when $P(A_1)P(A_2) = P(A \cup A_2)$ Now I've read another definition (albeit ...
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59 views

Law of large numbers: second moment tends to 0, but $S_n/n$ doesn't converge a.e.

What is an example of a sequence of random variables $\{X_n\}$ on a probability space $(\Omega, \mathscr{F}, P)$ such that $E(X_n^2) \to 0$ but it is not the case that $$ \frac{S_n - E(S_n)}{n} \to 0 ...
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54 views

Permutations of numbers

Given the five digits $1,3,4,6,$ and $7$. In the following question, it should be understood that repition of a digit is not allowed. (i) How many three-digit numbers can be formed from the ...
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137 views

Salesmen in a supermarket Poisson

We have a supermarket in which customer enter at Poisson rate 2. There are two salesmen near the door who offer passing customers samples of a new product. Each customer takes an exponential time time ...
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42 views

Are there “necessary” conditions for a solution to the multivariate, truncated Hausdorff moment problem?

I am looking for NECESSARY conditions for a solution to the multivariate, truncated Hausdorff moment problem (i.e., conditions under which a given finite sequence of numbers is the sequence of first ...