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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A basic question on dostribution of longitude and latitude

Let $\Theta$ and $\Phi$ be the longitude and latitude of a random point on the surface of the unit sphere in $\Bbb R^3$. I have to prove that $\Theta$ and $\Phi$ are independent, $\Theta$ is uniformly ...
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11 views

finding conditional expectation under binomial distribution.

Suppose X and Y independent and are both binomial random variables with parameter N, p Compute E(X|X+Y).
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19 views

weak convergence and composition

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued ...
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22 views

Is the martingale propertey preserved by taking weak$^*$-limits?

Let $(\Omega,\mathcal F)$ be a measurable space, $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that ...
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1answer
21 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
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23 views

Signed finite Radon measures with vague topology

If $X$ is a locally compact and $\sigma$-compact metric space. Let $M(X)$ be the space of signed finite Radon measures on $X$. (1) Show that measures with finite support is dense is $M(X)$ in the ...
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Singular distributions: Applications and Instances

This is the duplication of the question I asked here. I repeat it here with hope of getting new answers. Singular distributions are special mathematical objects. They have an interesting property ...
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32 views

A basic question on sequence of random variables

Suppose for a sequence of mean zero finite variance random variables $X_n$ the following is true for all $\epsilon \gt 0$ : $$ \lim_{n->\infty} \frac{\int_{|x| \geq \epsilon \sqrt{r_n} \sigma_n} x ...
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14 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?
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13 views

Show $W_{\infty}$ is not tail measurable but {$W_{\infty}=0$} is

The random walk: ({$\omega_{n}$},$\mathbb{P}$) simple random walk on d-dimensional integer lattice $\mathbb{Z}^{d}$ and the random environment: $\eta$={$\eta(n,x):n\in\mathbb{N}, x\in \mathbb{Z}^{d}$} ...
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39 views

Bounds for sum of random variables

Let $A_1,...,A_M$ be random variables, not necessarily independent. For each one of them I know that $P( A_i \geq a )\leq B_i, \quad i=1,2,...,M$. How can I retrieve lower/upper bounds for ...
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22 views

probability of covariance proof [duplicate]

I started with covariance formula and then I got stuck. I don't know how to do further steps Let $X$ denote a random variable and $Y = g(X)$ where $g$ is increasing (on $\operatorname{Ran}(X)$). Show ...
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1answer
16 views

Joint distribution and Integration

I was trying to prove a problem in my notes and now I need to whether prove or disprove the following claim: Assume $X,Y,W,Z$ are random variables defined on $(\Omega,\mathcal{F},P)$. If $(X,Y)$ and ...
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1answer
14 views

Showing a process is a martingale from Ito's lemma

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing ...
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20 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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21 views

Markov Chain Geometric Convergence

Perhaps due to my non-mathematician background (lack of Measure Theory knowledge), I have some difficulties about the Markov Chain Theory. Given an ergodic (irreducible and aperiodic) Markov Chain ...
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17 views

Formula for a Skewed Distribution

I'm trying to create a model of some data, using a skewed normal distribution. I have the following data: Mean Median Standard Deviation from the mean Standard Deviation from the median I've been ...
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37 views

A basic question on characteristic function

Suppose I have two random variables $X$ and $Y$ for which characteristic functions are same. Let $F$ and $G$ be their distribution functions. I have to prove that $F$ and $G$ have the same set of ...
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1answer
22 views

A basic question on measure

Suppose I have a measure in $B(\Bbb R)$ such that for each real number there is a neighbourhood where the measure is zero. Is that measure be necessarily zero measure ? How to prove it ? I can't take ...
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1answer
33 views

Why this random variable is uniformly distributed over the surface of the sphere

This is one of exercises in Probability: theory and examples, Durrett 3.2.15. Show that if $X_n = (X^1_n, ...,X^n_n) $ is uniformly distributed over the surface of the sphere of radius ...
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2answers
23 views

Probability of begin cut

Any athlete who fails the Enormous State University's women's soccer fitness test is automatically dropped from the team. Last year, Mona Header failed the test, but claimed that this was due to the ...
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18 views

Find linear line of data following gaussian distribution

I have a data that following gaussian distribution $x=\{x_1,x_2...x_n\}, p(x) ~is ~N(\mu,\sigma)$ Now, I want to find a linear line fitting all the data such as $y=ax+b$ The line subject to ...
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33 views

question about probability problem

How is the last column calculated? I don't understand, and I don't understand the explanation. $P(A \cap B)$ is calculated by $P(A)P(B\mid A),$ right? How is $P(A\mid B)$ calculated? Thanks
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1answer
37 views

Finding Random variables measurable

If [0,1] is our sample space and our sigma algebra is generated by all segments of the form [0,2^(-n)]. How can we describe the random variables measurable with respect to our sigma algebra? I'm ...
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21 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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46 views

Solve my probability doubt? [on hold]

A parent gives birth to two children. One of the child is surely a male, what is the probability of having both male child? Common answer 1/2 Actual answer 1/3
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Approximate the local behavior of an unknown distribution with uniform distribution.

consider an arbitrarily smooth distribution function over support $S$. I am only interested in local behavior that happens in a very small area. To what extent can I approximate the local distribution ...
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15 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
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28 views

Probability: Disease and Diagnosis

The probability of occurrence of a certain disease in a population is $1/101$. A diagnostic test has $9$ out of $10$ chances to detect the disease when the tested subject is actually affected. On the ...
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19 views

Conditional Probability Question - on route availability

Hey Guys I am seemingly stumped with this question I have gotten involving conditional probability and routes Suppose route $A$ to $B$ is available 0.5 of the time An alternative route to B from A ...
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47 views

Donsker for randomly stopped processes

A question regarding Donsker's invariance principle. Donsker states that if $X_1, X_2, ...$ are independent and identically distributed with mean $0$ and variance $\sigma^2$ and if $S_t^n$ is the ...
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1answer
20 views

Understanding Poisson Point Processes

I'm currently trying to understand PPPs. In the following I will state what I believe to know (please correct me if I'm wrong). I'm considering a PPP with intensity $\lambda$ on area $A = [-0.5, 0.5] ...
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23 views

A basic question expectation

Let $X$ and $Y$ be random variables respectively such that $E[X]=1$ and $E[Y] =0$ , $X^2 + Y^2=1, |X|\leq 1,|Y|\leq 1 $. Is it true that $X=1$ a.e. This has been used in some proof in Athreya Lahiri ...
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28 views

when is the product of averages greater than the average of products?

Has any theoretical work been done on when (and how often) the product of averages is greater than the average of products? That is, suppose $X(i, j)$ are positive real numbers for $i=1,\ldots,n$ and ...
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1answer
39 views

Inferring symmetry of a distribution from its marginals

Let $X=[X_1,\ldots,X_n]$ be a continuous random vector of size $n$ with density function $f_X(x_1,\ldots,x_n)$. If all the marginals \begin{align*} \int \ldots \int f_X(x_1,\ldots,x_n)\, ...
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22 views

indepence transitive property?

For the events A and B are independent and B and C are independent is A and C independent I used coin tosses to try to model this with A = H B = T and C = H in seperate fair tosses I get that they ...
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30 views

$E[\hat{\theta}_{MME}] = E[\frac{1- 2\overline{y}}{\overline{y}-1}] = \int_0^1 \frac{1- 2\overline{y}}{\overline{y}-1}(\theta+1)y^\theta dy$..?

Let $Y_1, Y_2,\dots , Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
31 views

Gaussian vectors and covariance matrix.

The following is a part of a question I was given in stochastic processes course. It goes like this - I am given a series of gaussian iid random variables $\{V_i\}_{i=1}^N$ , the variable $X_0 \sim ...
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9 views

What is the relationship of the EMD (Earth movers Distance) and total variation (and other probability measures)?

I was trying to understand different methods for comparing probability distribution and saw the following paper/reference: http://arxiv.org/abs/math/0209021 In it it defines and compares and ...
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1answer
15 views

Variance formula - need verifying [on hold]

$$D^2(X)=D^2(E(X|Y))+E(D^2(X|Y))$$ Can someone verify that this is true and why?
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2answers
19 views

What does the sup function mean in the context of metrics for probability measures/distances/differences?

I was studying different probability metrics and distances and came across the following source: ...
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24 views

Stochastic domination by coupling

The following is a slightly streamlined version of Exercise 7.5 in Dubashi & Panconesi's "Concentration of Measure for the Analysis of Random Algorithms": Let $X$ and $Z$ be independent random ...
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1answer
26 views

Expected value vs values which happen with the biggest probability

If $X$ is a random variable from binomial distribution $Bin(n,p)$, then $$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$ where $p$ is the probability of one success. The expected value of random variable ...
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46 views

Convergence in probability implies convergence in mean under one additional condition

Prove that if random variables $X_n$ are dominated by an integrable random variable then $E[X_n] \to E[X]$ follows if $X_n$ converges to $X$ in probability. Hint: Use the following theorem : A ...
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34 views

$1/(1+X_n)$ bounded in probability

I am trying to prove that if $X_n\rightarrow 0$ in probability, then $1/(1+X_n)$ is bounded in probability. My attempt is: $$P(\frac{1}{1+X_n}<\frac{1}{1-\epsilon})=P(|1+X_n|>1-\epsilon)\\ \geq ...
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35 views

A basic question on convergence in prob. and a.s. convergence

If random variables $X_n$ are dominated by an integrable random variable then $E[X_n] -> E[X]$ follows if $X_n$ converges to $X$ in probability. Proof: Take any subsequence $\{E[X_{n_k}]\}$. I ...
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33 views

Limit of ratio of random sequences

Let $X_1, X_2, \dots $ i.i.d random variables with the following properties. (1) $\mathbb{P}(|X_j|>x)= x^{-\alpha}L(x)$, where $\alpha \in (0,1)$ and $L(x)$ is a slowly varying function. (2) ...
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84 views

Compare two estimators by using the their Expected value and variances

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
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1answer
78 views

Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distrib… Help find finding $ \text{Var}\left[\hat{\theta}_{2}\right]$

Let $Y_1, Y_2,\ldots,Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Find the efficiency of $θ^1$ relative ...
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Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function with density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...