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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How to prove the uniqueness of probability measure

Probability essentials P-21 Theorem 4.1 (b) Let $(p_\omega)_{\omega \in \Omega}$ be a family of real numbers indexed by the finite or countable set $\Omega$. Then there exists a unique probability ...
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15 views

Proving that a random variable is stable

If $\overline{F} = 1- F_X$ where $F_X$ is a cumulative distribution function of $X$. Then if the following is satisfied: $$ \lim_{x \to \infty} \frac{\overline{F}(\lambda x)}{\overline{F}( x)}= ...
3
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1answer
57 views

rolling a single die ten times

I have the following problem on a homework assignment for my Probability theory course: You roll a single six sided die ten times. What is the probability that you roll four 1's, three 2's, and three ...
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31 views

Conditional distribution

One point is chosen at random in the square $Q=\{|x| + |y| \leq 1\}$. Let $(X, Y)$ coordinates that point. a) The random variable $X$ and $Y$ are independent ? b) Find the density of $X$ given that ...
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31 views

Understanding the geometric distribution

Simple question that has to do with the interpretation of the geometric distribution and frequency function: $P (X=k) = (1-p)^{k-1}p $ for $k = 1,2,3... $ where we are interpreting X as being up to ...
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1answer
32 views

Probability that a stochastic process is below a special random level

Given a stochastic process $x(t)$ over time $t \in [0,T]$, and a given (deterministic) $\tau$, where $0<\tau<T$, define a random variable $x^{*}$ as $$ x^{*} \triangleq \inf\bigg\{y: ...
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1answer
19 views

Conditionally independent and intersection

I'm trying to show that, given events $A,B,C,D$, such that $A,B$ are conditionally independent given $C$, whether or not $A,B$ are conditionally independent given $C\cap D$. I spent a couple of hours ...
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21 views

Help with a definition involving multiple suprema/infima

I have trouble understanding a definition that comes up in a proof of the Prokhorov theorem. Let $E$ be a Polish space and $M$ a set of probability measures on the Borel $\sigma$-algebra on $E$. From ...
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25 views

Independence of Events in Lovasz Local Lemma

Let $G$ be a (finite) graph with maximum degree $d$ and vertices $v_{1}, \dotsc ,v_{n}$. Let us associate an event $A_i$ with $v_i (i = 1, . . . , n)$ and suppose that $A_i$ is independent of the ...
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2answers
49 views

Is it right to give equal chances?

I have a doubt ! Problem is : In a certain town, the probability that it will rain in the afternoon is known to be $0.6$ Moreover, meteorological data indicates that if the temperature at noon ...
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15 views

Laplace vs Fourier density representation of a positive rv

Given a general random variable $X$ with density function $f(x)$ and characteristic function $\phi_X(u)$ we can go back and forth between the density and the characteristic by using the Fourier ...
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1answer
21 views

A conditioned on B is independent of C

Let $A,B,C$ be measurable sets on a probability space. I'm trying to understand the meaning of the sentence: A conditioned on B is independent of C. The conditional probability was defined as: $$ ...
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25 views

Application of Riesz-Markov-Kakutani representation

Let A be finite set, $\{f_i\}_{i=1}^{n}:A\to \mathbb{R}$ non-negative and with the following property for every $\sum \lambda_{i}=1$: for any $g=\sum \lambda_{i}f_i$ there exists $a\in A$ s.t. ...
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37 views

Generating structure of Borel field

On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the author wrote: ...and there are Borel sets that cannot be arrived at from the intervals by any finite sequence of set-theoretic ...
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24 views

Example of a bounded simple process $A_t$ that changes value only once s.t. $\int_0^t A_s dB_s$ doesn't have normal distribution? [closed]

As the title of the question suggests, what is an example of a bounded simple process $A_t$ that changes value only once such that$$\int_0^t A_s\,dB_s$$does not have a normal distribution?
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1answer
45 views

A coin probability question

Let $p$, $q$ be values in $[0,1]$ and $\alpha \in [0,1]$. Assume $\alpha$ and $q$ known, and that $p$ is unknown parameter we would like to estimate. A coin is tossed n times, resulting in the ...
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Hypothesis Testing with given Pearson correlation.

There are two independent sets of samples , female and male. The problem is to calculate the 95 % C.I. of the mean of total( male+ female, two samples) population. -> Female : n=100, mean= 169.1, sd ...
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2answers
25 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
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41 views

Uniform convergence of characteristic functions implies uniform convergence of distribution

Let $F(x)$ and $(F_{n})_{n\geq 1}$ be some distribution functions and let $\varphi(t)$ and $(\varphi_{n})_{n\geq 1}$ be their respective characteristic functions. I am trying to show that if: $\sup_t ...
2
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0answers
48 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
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25 views

Uniform integrability in central limit theorem

Suppose $X_1,X_2,\ldots$ are i.i.d. with $P(X_1=+1) = P(X_1=-1) = \frac 12.$ We know that $n^{-1/2}\sum_{i=1}^n X_i \stackrel{d}{\to} Z$ where $Z\sim\mathcal{N}(0,1).$ How steeply can a continuous ...
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24 views

Converting weak convergence to convergence in probability

We know that if $X_n\to X$ in distribution, it need not be the case that $X_n\to X$ in probability, the classic example being the central limit theorem. Suppose we are given a sequence of ...
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132 views

The mathematics of a drinking game called Spoon

This question is about the drinking game Spoon. It was asked on reddit.com/r/math : http://www.reddit.com/r/math/comments/3i9790/drinking_game_turned_mathematical/ The question is whether the person ...
2
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1answer
28 views

Poisson Approximation of Binomial

I have to prove the Poisson approximation of the Binomial distribution using generating functions and have outlined my proof here. Given, \begin{align} & \lim_{n\to \infty} np_n = \lambda \\ ...
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47 views

Sufficient condition for convergence in distribution in the plane

I'm trying to show convergence in distribution for a sequence $X_n$ of random variables in the plane. Here's what I know. I have a sort of squeeze theorem for the probability of the r.v.s being in a ...
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1answer
39 views

Does it mean “two successive tosses is the same” is same as “two successive tosses is either heads or tells”?

I got confusion ! Does it mean "two successive tosses is the same" is same as "two successive tosses is either heads or tells" ? I have two problems : Q. I have an unbiased coin , assuming ...
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0answers
14 views

Probability of Detection and pulse-pulse decorrelation time [migrated]

Not sure if this is the right forum, but thought I'd ask anyway: I'm trying to analyze the probability of detection ($P_d$) for a Swerling II target. I know that the Swerling II target assumes that ...
2
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1answer
32 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
2
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1answer
29 views

Constructing dependent sequences of random variables

It is easy, given some random variable $X \colon \Omega \to \mathbb{R}$ on a probability space $(\Omega, \mathbb{P})$, to construct an i.i.d. sequence $X_1, X_2, \ldots$ distributed as the law of $X$. ...
2
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1answer
33 views

Show that is a probability space

Let $ \Omega:= \{(x,y) \in \mathbb{R^2}:0<x,y \leq 1 \}$, let $\mathcal{F}$ be the collection of sets of $\Omega$ such that $$ \mathcal{F}:= \{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \}$$ ...
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1answer
37 views

'Markovian Property' vs 'Memoryless Property'

The two properties have the commonality in the sense that they predict the future based on the current state, not on the whole history of how the process wandered into the state. Then, what is the ...
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33 views

Show that $S_n/n$ converges almost surely if $S_{2^n}/2^n$ converges almost surely

Let $X_n$ be independent random variables such that $\dfrac{S_n}{n}\to0$ in probability and $\dfrac{S_{2^n}}{2^n}\to0$ almost surely. Show that, $\dfrac{S_n}{n}\to0$ almost surely. Here ...
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1answer
29 views

Can this probability be shown by using the properties of Lebesgue integration

(Grimmett and Stirzaker - Probability and Random Processes - Exercise 1.3.5) I am studying Lebesgue integration in parallel to probability theory, and my question is: Can the following be shown by ...
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23 views

Show that the likelihood ratio converges to $0$ a.s.

Let $S$ be a finite set, for simplicity assume $S=\{1,2,...,m\}$. Let $f_0$ and $f_1$ be two non-equal probabilities defined on $S$, with $f_0(j)=P_0(X=j)$ and $f_1(j)=P_1(X=j)$, such that ...
2
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1answer
27 views

measurability of a sum of random variables

Let $\{X_n\}_{n = 1}$ be a sequence of random variables. Let $Y_n = \sum_{i=1}^n X_i$ Then, I want to show that $\sigma(Y_1,Y_2,\dots,Y_n) = \sigma(X_1,X_2,\dots,X_n)$ for every $n$. It is clear to ...
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1answer
19 views

multivariate convergence in law

Suppose $X_n \overset{\mathscr{L}}{\longrightarrow} X$, and $Y$ is another random variable which may be depending on $X_n$. Then it seems not true that we have the following joint convergence in law ...
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1answer
22 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
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54 views

If $X$ is such that $c P(X\geq c) \leq E(Y; X\geq c)$ for every $c$ then so is $X\wedge n$

I came across the following as I was reading the book Probability with Martingales by Williams (pg 143): Assumption: $X$ and $Y$ are non-negative random variables such that $$c P(X\geq c) \leq ...
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2answers
41 views

Resampling operation

I am reading from an arXiv.org paper the following math text: "Let $x\in \{−1, 1\}^I$ be random and uniform, and let $y$ be obtained from $x$ by resampling each coordinate with probability ...
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37 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
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1answer
18 views

If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$

Let $\{X_k\}_{k=1}^n$ be iid random variables that are symmetric around $0$ i.e. $X=-X$ in distribution. Define $S_n=\sum_{i=1}^nX_i$. Then show $P(|S_n|\geq\max_{1\leq i\leq ...
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1answer
18 views

Show that $S_n$ converges almost surely to $\infty$

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let ...
2
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2answers
51 views

piecewise weak convergence in $C[0,1]$

Let $P$ and the sequence $P_n$ be probability measures on $C[0,1]$ with the uniform metric. Fix $0<u<1$ and let $\Pi_1$ and $\Pi_2$ be the projections from $C[0,1]$ onto $C[0,u]$ and $C[u,1]$, ...
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1answer
21 views

Measurability of a version of a random variable

If $X$ is a ($\mathcal{F}$-measurable) random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and $Y$ is a version of $X$ in the sense that $\mathbf{P}(X \ne Y) = 0$ and ...
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0answers
35 views

Zero conditional entropy

This question is related to this math.se question but I need a bit more guidance. For two discrete random variables $X,Y$ we define their conditional entropy to be $$H(X|Y) = -\sum_{y \in Y} Pr[Y = ...
2
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2answers
44 views

expected values of identically distributed random variables

Let $X$ and $Y$ be identically distributed random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then, if I let $F_X$ and $F_Y$ denote the distribution functions of $X$ and ...
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43 views

Every random variable $X$ can be written as $X=\lambda Z_1+(1-\lambda)Z_2$, for $Z_1$ discrete and $Z_2$ continuous random variables.

Show that every random variable $X$ can be written as $$X=\lambda Z_1+(1-\lambda)Z_2$$ for a discrete random variable $Z_1$, a continuous random variable $Z_2$, and a real value $\lambda$. This ...
2
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1answer
54 views

Taylor expansion of characteristic function in probability theory

In probability theory, what is the Taylor expansion of characteristic function? I know this is a basic question but I couldn't find a full answer.
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1answer
26 views

Does the following result require the random variables to be independent?

I am sitting with the book Labelled Markov Processes by Prakash Panangaden, and on page 79 he defines what it means for a set of random variables on a probability space to be independent, and after ...
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2answers
51 views

Which one of the following versions of Bayes' theorem is correct?

I've seen two versions of Bayes' theorem: I've seen this very long version from a frequentist probability class: $$ P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B^c)P(B^c)} $$ where $B^c$ is the event ...