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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121 views

Sum of Dependent random variables pdf alternative?

I'm attempting to calculate mutual information for two cases and running into some difficulty. Let me describe the setup: $X\sim$ independent and identically distributed symbols $(+1/-1) w/ p=0.5$ ...
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81 views

Inverse Sampling Method

I want to prove from first principles that if $X$ is a continuous random variable with cdf $F$, then $Y = F(X)$ is uniformly distributed on $[0,1]$. For $y \in (0,1)$, if we define $ a = \sup\{ x: ...
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35 views

Bound for the uniform distance between two Gaussian CDFs

Consider two $p$-dimensional Gaussian variables $X$ and $Y$ with mean zero (for simplicity) and covariance matrices $\Sigma$ and $\Gamma$, respectively. With $F$ and $G$ denoting the $p$-dimensional ...
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26 views

ML Estimation, graphical method?

I have a problem in statistics that I don't quite know how to do: "Generate a 1000-element data sample from the Rayleigh distribution. Graph the log-likelihood function $\ln L(\alpha; \vec{x})$ as a ...
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41 views

Small time convergence of Ito diffusion

Let $X_t$ be such that $dX_t=\alpha(X_t)dt+\gamma(X_t)dW_t,\,X_0=x_0$, and let $f(x)$ be a bounded, positive, continuous function. If $f$ has two bounded derivatives in a neighborhood of $x_0$, then ...
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70 views

Interchanging limits and expectations

Here is a problem that I have been trying to solve for some time: $\lim_{N \rightarrow \infty} \frac{1}{N}E[\sum_{i=1}^{N} f(x_i)]$. Now the pdf of $(x_1,x_2,...,x_N)$ is given as: ...
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28 views

Sample space construction in general case

Is there any general algorithm of constructing Sample Space from a given experiment? This question might seem trivial, but it was causing me some confusion before realizing that objects that are used ...
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69 views

Multistage Diffusion

I am trying to find the probability of a particle being in $R$ at time $t$, under the assumption the particles undergoes transition between two diffusion processes (characterized by two different ...
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34 views

Interchange of partial differentiation for a CDF

I have a question regarding the interchange of the order of partial differentiation of a cdf $F(x,y)$, say. is it always correct that (provided $F$ admits partial derivatives) that $$ ...
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47 views

Is it possible to have $\mathbb E S_n \to \infty$ but $\inf S_n = -\infty$ a.s.?

The original question is: Let $X_1, X_2, \ldots$ be i.i.d. with $\mathbb P (X_1 = 1) = p > 1/2$ and $\mathbb P (X_1 = −1) = 1 − p$, and let $S_n = X_1 + \cdots + X_n$. Let $\alpha = \inf\{m : ...
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76 views

No subgraph with average degree greater than $3c$ for $G(n,m)$ where $m=cn/2$

Consider the graph $G_{n,m}$, which has $n$ nodes and $m$ random edges. Let $m=cn/2$ where $c>0$ is a constant. I want to prove that with high probability (i.e. with probability $1-o(1)$), there is ...
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43 views

When is the Fourier Transform Cyclic?

This is kind of a continuation of Questions about stable laws but I did not want the subject to be too broad in one question. The only stable laws that are also symmetric and nondegenerate (and hence ...
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25 views

Questions about stable laws

For the duration of this post, $\log$ will mean the logarithm of a curve, and not any particular branch of the complex $\log$. So for a continuous curve avoiding $0$ with initial value $1$, there ...
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27 views

Question on MRF with Efficient Approximations [Boykov98]

My question is about this paper. Although most of the explanations are perfectly clear to me, there is this one part I feel unsure about.. If you could enlighten me, I would really appreciate it. ...
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43 views

Limit of a continuous diffusion process

Somewhere in my ancient set of notes it says that for a continuous diffusion process, for every $\epsilon>0$, $$\lim_{t\to 0}\frac{\mathbb{P}(|X_t-X_0|>\epsilon)}{t}=0.$$ Can this be extended to ...
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73 views

a.s. convergence of exponential function

Suppose $Ω=[0,1]$, $\mathcal F=\mathcal B\left([0,1]\right)$, $\mathbb P=\lambda$, and consider $$X_n=e^n\chi_{A_n}, \qquad A_n=\left[0,\frac 1n\right].$$ I'd like to show that the sequence ...
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33 views

sampling schemes for binomial distribution

Two acceptance sampling schemes, A and B, are proposed for deciding whether or not to accept a large batch of items from a production process in which 5% of the items produced are defective. Scheme A: ...
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32 views

Proof that the radius is a sufficient statistic for a circle

How can I prove that the radius of a circle is the sufficient statistic for the probability of choosing random points in the area of the cricle?
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35 views

Calculating the probabilities of different lengths of repetitions of numbers of length 6

This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4 except now I'm having problem with numbers of length 6. ...
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44 views

If the fields $F_\alpha^0$ are independent, then so are the B.F.'s $F_\alpha$.

Problem 4 of Section 3.3 from "A Course in Probability Theory" by Kai Lai Chung Fields or B.F.'s $F_\alpha(\subset F)$ of any family are said to be independent iff any collection of events, one from ...
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45 views

Orthogonalization of a set of random vectors

Suppose $w_1$ and $w_2$ are zero-mean jointly Gaussian random vectors. Further suppose that they have a covariance matrix given by $$ \mathbf{cov}\begin{bmatrix}w_1 \\ w_2\end{bmatrix} = ...
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57 views

What does commensurable in this context mean?

Suppose that we observe a finite set $T\subseteq U$ where $U$ can be both finite or infinite. Further we assume that a proability measure $P_T$ is defined on all the subsets $S\subseteq T$.  Then by ...
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63 views

Intregral of exponential of Shannon Entropy Function

Here I am going to ask a similar question as rde asked , that is what is the integral of exponential of entropy function. That is what is the value of $F[H(x)]=\int_{-1}^{+1} e^{ikH(f(x^2))} dx$ ...
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47 views

Does asymptotic normality imply convergenc in probability

Is that true that if $X_n$ is $AN(u,\sigma_n)$ ,then $X_n$ converges to $u$ in probability if and only if $\sigma_n \rightarrow,n\rightarrow\infty$? If it is true , then how to prove it ? Thanks in ...
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93 views

Facts about Brownian motions and Markov chains on concrete constructions

Often times, the approach to prove facts about Brownian Motions and Markov Chains which are continuous and discretely indexed stochastic processes is to construct a special space. The proof of ...
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25 views

Submartingale bounds

Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
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53 views

Probability that values from different normal distributions will be in a certain order?

I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that they are in a certain order. For example, I have ...
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55 views

Simple symmetric random walk - is my assumption correct?

Let $a,k\in\mathbb{N}$ with $a<k$, and let $(S_n)$ denote a simple symmetric random walk starting at $a$. We know that $(S_n)$ is a martingale, of course. Consider the stopping time $\tau=\inf\{ ...
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44 views

Showing that $|X|^p$ is a submartingale

I am trying to get my head around something, and I'm pretty sure the answer is lurking right in front of me. We're in $(\Omega,\mathcal{F},P,(\mathcal{F}_n))$, and we denote by $(X_n)$ a ...
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75 views

How to compute $p(\pi>\theta_2+N>\theta_1+\pi, \hspace{1em}\theta_1<0, \hspace{1em}\theta_2<\theta_1+\pi)$ for dependent jointly Normal RVs

How would one go about computing: $P(\pi>\theta_2+N_1>\theta_1+\pi, \hspace{1em}\theta_1<0, \hspace{1em}\theta_2<\theta_1+\pi)$ given $\theta_1, \theta_2$ are jointly Normal (mean=0, ...
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49 views

Why is K-L divergence defined as it is?

Why is the K-L divergence defined this way: if $P$ and $Q$ are probability measures over a set $X$, and $P$ is absolutely continuous with respect to $Q$, then the Kullback–Leibler divergence from ...
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32 views

When a family of measures provide continuity?

Consider a mapping $m: \mathcal{B}(X) \times P \rightarrow [0,1]$, where $X \subseteq \mathbb{R}^n$, $P \subseteq \mathbb{R}^m$, and $\mathcal{B}(X)$ denotes the Borel sets. $\forall p \in P$, ...
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63 views

Power Curves from Normal Distribution.

The following is a homework problem that I cannot figure out because I am having trouble finding the Type II error. Construct a power curve for the $\alpha = 0.05$ test of $H_0:\mu = 60$ versus $H_1: ...
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57 views

something about property of Bernoulli random variables

Let $b_i, i=1, \ldots, n$ be Bernoulli random variables with probability $P(b_i=1)=2k/n,$ where $k\leq n.$ Show the following: Let $\chi$ be an indicator function that $k$ out of $n$ of $b_i$ are ...
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60 views

Relation between convergence of pmfs, in distribution and in total variation for discrete distributions?

For a sequence of discrete probability distributions, are following convergence modes equivalent: pointwise convergence of their pmfs, convergence in distribution (assuming the distributions are ...
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142 views

Distribution of convex combination of i.i.d Gamma random variables

I am wondering what one can say regarding the convex combination of i.i.d Gamma random variables? Specifically, consider $x_{i}$ be $Gamma(\theta,1)$, then would we have the following and if yes, ...
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62 views

Which probability distribution does my discrete variable approximate?

The discrete random variable X is a count variable of 45 Bernoulli trials and can take on values from 0 to 45. The mean is 2.34 and the variance is 18.39. The average probability of scoring 1 in a ...
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38 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
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57 views

Probability computation $P(X_n/\log(n))$

Let $X_1, X_2, ...$ denotes a sequence of i.i.d. random variables such that $X_1$ ~ $exp(1)$ and c>0. What is $P( X_n/\log(n) > c$ for infinitely many $n$'s) ? Can I simply say that $P(X_n > c ...
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82 views

Mathematical notation for probability trees and their usage

A commonly used tool for visualising and solving Conditional Probability problems is the tree diagram of events and their associated probabilities. (Tree Diagram). How can one represent particular ...
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212 views

Random Variable on a Sphere

Not sure where to start with this problem: For any $d\geq 1$, we admit that there is only one probability measure $\mu$ on $\mathcal S_d$, (the $(d-1)-th$ dimensional sphere embedded in $\mathbb ...
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106 views

A generalization of the central limit theorem

Let $\{X_n\}$ be a uniformly bounded family of independent mean zero random variables. Suppose each $X_n$ has variance $\sigma_n^2$, and that $\sum_{n=1}^\infty \sigma_n^2 = \infty$. I am trying to ...
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483 views

Hypothesis Testing on Exponential distributions

Let $X_1, \dots, X_n$ be independent exponential $(\theta)$ random variables. Suppose we are interested in testing $H_0: \theta = \theta_0 = 1$ versus $H_A: \theta = \theta_1>1$. Consider two tests ...
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429 views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
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37 views

Lebesgue-Stiletjes measure on $\mathbb{R}^2$

Let $F$ be a two variable continous function that satisfy \begin{equation} F(x_1,y) \leq F(x_2,y) \text{ for } x_1\leq x_2, \end{equation} \begin{equation} F(x,y_1) \leq F(x,y_2) \text{ for } y_1\leq ...
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286 views

The mean and variance of the random variable with Rician distribution

What is the mean and variance of $Z_0=\frac{\sum\nolimits_{i = 1}^{{n}}R_iZ_i}{\sum\nolimits_{i = 1}^{{n}}R_i}$, where $Z_i$ is a constant and $R_{i} =\sqrt{X_i^2+Y_i^2} $ ${(i=1,\ldots ,n)}$? $X_i$ ...
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84 views

What does $E_{x}[f(x,y)]$ mean?

Frankly, the title says it all. In the context of expectations in probability, what does it mean $$E_{x}[f(x,y)]$$ In the single variable case, $E[f(x)]$ is equal to $\displaystyle \int p(x)f(x)dx$, ...
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81 views

How to prove this inequality involving integration with respect to Brownian motion?

If $B_t$ is the Brownian Motion, I have to verify that $$E\left\lvert\int_s^t G(t,w)\,dB_t\right\rvert^6\leq 15^2\cdot (t-s)^2\cdot\int_s^t E\lvert G(t,w)\rvert^6\,dt$$
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53 views

Deriving the process of successfully consumed requests from the process of request-producers and the process of request-consumers

The title is not very straightforward I understand, but you will soon realize it was not so simple to describe in few words this problem. The problem Consider a system consisting of: A process of ...
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25 views

Levy process: reversed $\overset{d}{=}$ dual : $X_{t-s}-X_t \overset{d}{=} - X_s$

Let $X$ be a Levy process, there is a theorem which says $$ X_{(t-s)^-} - X_t \overset{d}{=} - X_s, $$ for $0 \le s \le t$, $t$ fixed. How do we proove this without a graphical argument? In ...