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 to calculate this CDF?

Let suppose that we have three points in the euclidean plan $\mathbb{R}^2$ which are depicted inside a circle of radius $R$ as follow: $P_1=(D,0)$ (the center of the circle), $P_2=(0,0)$, and ...
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33 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
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30 views

Interchanging limits in stochastic order notation definition

A sequence of real-valued random variables $(X_n)$ is said to be $O_P(b_n)$ where $(b_n)$ is a sequence of positive numbers if $$ \lim_{T\to\infty} \limsup_{n\to\infty}P\{|X_n|>T b_n\}=0$$ Is it ...
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20 views

Expectation of O_p(1) process

Suppose $\{X_n \}$ is bounded in probability, i.e. $Prob(|X_n| > M_\epsilon) = \epsilon$ for all $n > N_\epsilon$, $M_\epsilon < \infty$. Is there any condition(s) to guarantee that ...
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34 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
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54 views

Question about computing the sample mean and variance values from a sample coming from a Weibull Distribution …

Let's suppose that I have a random sample x from a Weibul distribution with shape parameter k=1 and scale parameter λ=2... How am I supposed to compute the mean value of the sample ? Also what can I ...
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25 views

Probability using sup

I have to prove that $$P(A\cup B)=P(A)+ P(B)$$ if $A\cap B=\emptyset$. Using the definition of sup $$P(A)=\sup \left\{\sum_{x_n\in J}p_n:J\subset A, J\ finite\right\}$$ But i really don't know how to ...
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38 views

Conditional expectation of a continuous random variable

Let $\left( {X,Z} \right)$ be a continuous random vector with the PDF ${f_{X,Z}}\left( {x,z} \right)$ and ${f_X},{f_Z}$ corresponding marginal PDFs. Let ${f_{X|Z}}\left( {x|z} \right) = {1_{\left\{ ...
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43 views

Proof of “changes of sign” in one-dimensional random walk model [Feller's section 3.5, page 84]

Consider the one-dimensional random walk of a particle. We shall denote the individual steps by $X_1, X_2, \cdots$ with $X_i = \pm 1$ and the positions by $S_1, S_2, \cdots$ with $S_i = X_1 + X_2 + ...
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28 views

How to represent?

You are a well-known hedge fund manager in Wall Street circles. One of your wealthy clients has $\$1$ million dollars to invest in XYZ stocks. Currently, XYZ stocks are trading at $\$2$ per share. You ...
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27 views

Girsanov's theorem for OU process

Say that I've got the process $dr_t=a(b-r_t)dt+\sigma dB_t$ and that I want to calculate $E[\exp(-\int_0^T r_s ds)]$ by using Girsanov's theorem. How do I do this? I cannot seem to find an explicit ...
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38 views

Integrating a function of a random variable; $\int g(X) dP$

Assume a random variable $X$ on probability space $\Omega$, taking values in $\mathbb{R}$ with some known distribution $F(dX)$. Assume also a function of the random variable, $g(X)$. Does then the ...
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23 views

Random variables with random indexes

Suppose I have a pair of stochastic processes - $(X_n,I_n)$. Say $I_n$ is defined on $\mathbb{N}$. Both $I_n$ and $X_n$ may be Markov processes or iid random variables. I want to investigate the ...
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19 views

factoring series of multiplication

Given that $y = \frac{1}{N} \sum_{i=1}^{N}\log_2(1+|x_i|^2 y z). $ I simplify the equation into $2^y= \frac{1}{N}\prod\limits_{i=1}^{N}\left(1+x_i^2(y)(z)\right)$ ...
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23 views

Reflected borel sets are worse traps for Brownian paths.

This is for a research project. I am trying to prove that given a borel set, it's reflected version will have a lower Wiener measure of brownian paths intersecting it. In this paper they elaborate ...
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36 views

Relations between stochastic Big O-Notation, limsup and lim?

I have to solve a list of exercises on probability theory but I'm having several problems in understanding the following questions; could you help me? Thank you! Let $||.||$ indicate the Euclidean ...
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94 views

applying iterated expectation when conditioning on multiple random variables

The law of iterated expectations tells us that ${\bf E}\big [{\bf E}[X\, |\, Y]\big ]={\bf E}[X]$. Suppose that we want apply this law in a conditional universe, given another random variable $Z$, in ...
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43 views

find number of elements combinations covered by a given set of element groups

Here is the problem I have faced recently that I cannot deal with yet and I need some help: Given is the list of elements (numbered): e.g. [1,2,3,4,6,7,8] the count and size of groups, which can ...
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17 views

A basic question on weak convergence

I have to give an example of a sequence of distributions $\mu_{n} => \mu$ (weakly sense) but $\mu_n(A) -> \mu(A)$ fails for some $A$ and all the distributions $\mu_n$ and $\mu$ are coming from ...
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37 views

Dependence between a joint probability distribution

Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal ...
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60 views

Median-median inequality

An elementary result from Chebyshev's theorem is that the median and mean of a random variable do not differ by more than one standard deviation. I'm curious if there is a similar result for ...
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18 views

Determine the value of constant $C_n$ such that $(p_k,k=1,…,n)$ is a probability measure on $\{1,…,n\}$

For $\theta\in[0,1]$ and $n\ge 2$, define a sequence $(p_k, k=1,...n)$ by $p_k= \begin{cases} C_n(1-\theta)\min(n,n-k) & if\ k=1,...,n-1\\ \theta & if\ k=n \end{cases}$ ...
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12 views

Marginal Likelihood Prior.

I have a model with probability matrix for a distribution of $x,y={0,1}, p(x,y\mid w)$ where $w=[w_1,w_2,w_3,w_4]$ $$p(x=0,y=0)=w_1, p(x=0,y=1)=w_2$$ $$p(x=1,y=0)=w_3, p(x=1,y=1)=(1−w_1−w_2−w_3)$$ ...
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25 views

Calculation of Maximum A Posteriori

Give IID Data Samples $X_n = $ {$x_1, x_2, ..., x_n$} generated from a uniform distribution $U(x|0,\theta)$. $p(x|\theta) = U(x|0,θ) = ${$ \frac{1}{\theta}$ for $0 \leq x \leq θ$ and $0$ otherwise}. ...
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67 views

Expected waiting time of the process in the queue given that the process is served

Consider the following situation: There is one server with exponential service time with parameter $\lambda$. One process is waiting in the queue. The waiting time is exponential with parameter ...
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87 views

Bayes' Theorem Question, with a twist

I have a very old past high school exam question I am trying to solve (for interest only). It's a straightforward application of Bayes' Theorem, with the last part of the question containing a slight ...
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20 views

Two definitions of Convergence in distribution (random vector version)

I have seen several posts around about the two definitions for "convergence in distribution", but I was not sure whether the equivalence holds in the random vector version. The two definitions are: ...
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16 views

Calculating the Fisher matrix for a sample of a binomial distribution, probability dependent on underlying differential equations.

I am trying to calculate the Fisher matrix for an experiment in which the underlying parameters determining the growth of 2 bacterial populations are estimated via sampling the population ...
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24 views

Looking for a distribution where the scale amd sum lead to a closed form distribution

As the title says, say I have a finite number of i.i.d. random variables with positive support $X_1,X_2\dots,X_n$. I'm interested in finding a closed form expression for $S$, where: $S=\sum a_i X_i$ ...
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39 views

Counting problem about sub-matrices

EDIT (along the lines suggested by @sea turtles): Given a prime power $q$ and a positive integer $x\gt q$, how many subsets $A\subseteq{\bf F}_2^q$ have size $x$ and contain a subset $B\subseteq A$ ...
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24 views

Conditioning on $\mathcal{F}_\sigma$ for $\sigma$ stopping time

I'm trying to show that $E[E[\ \cdot\mid \mathcal{F}_\sigma]\mid\mathcal{F}_\tau]=E[E[\ \cdot\mid \mathcal{F}_\tau]\mid\mathcal{F}_\sigma]$ for stopping times $\sigma$ and $\tau$, I've come to the ...
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29 views

Subadditivity of events

I do not get this exercise: Let $A_i \in \mathbb{A}$ be a sequence of events. Show that: $P(\cup^{n}_{i=1} A_i) \leq \sum^n_{i=1} P(A_i)$ The solution is: Set $B_1 = A_1$, $B_i = A_i \setminus \ ( ...
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43 views

Change of variables in Lebesgue-Stieltjes integral

Suppose $F$ is a probability distribution function with corresponding measure $\mu_F$ on $\mathbb{R}$. Suppose moreover that $f:\mathbb{R} \rightarrow\mathbb{R}$ is a continuous, increasing function. ...
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16 views

Calculating the joint distribution of an affine stochastic process

I have a recursively defined system given by $$X_i = X_{i-1}H_i+N,$$ where $H_i$s are i.i.d. exponential random variables and N is a constant. At the $n$th iteration I have $$X_n = ...
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38 views

Please check the question: Compute $EX$

Question: A box contains $10$ balls numbered $1,2,\ldots,10$. A random sample of $7$ balls is selected. $X=$ the smallest of the numbers drawn. Compute $E(X)$ $R(X)= \{1, 2, 3, 4\}$ ...
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27 views

Normal Approximation for a Binomial

A blackjack player at a Las Vegas casino learned that the house will provide a free room if play is for four hours at an average bet of 50 bucks. The player's strategy provides a probability of .49 of ...
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35 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
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35 views

Why define shift invariant set as $\Lambda=\tau^{-1}\Lambda$?

Can we define shift invariant set as $\Lambda = \tau\Lambda$ instead of $\Lambda = \tau^{-1}\Lambda$, where $\tau$ is the shift operator? Can permutable set be defined as either $\Lambda = \pi ...
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13 views

Sum of random variables for 2m tape

we use 2 metre tape for distance measurement and that the measurement error for the full tape length has 0 mean and variance 1.5cm^2. Find the mean and the variance if the total distance measured by ...
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43 views

The relationship of $\sigma(f(X))$ and $X$

If X is a random variable and f is a measurable function, 1) Is f(X) measurable with $\sigma(X)$ ? 2) Is X measurable with $\sigma(f(X))$ ? Please give proof & example or counter example. Ok ...
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36 views

An outer meassure not being probability measure

I have to prove that an outer measure is not necessary a probability measure. I have this example: Let $\Omega $ be infinity, for every $A \subset P(\Omega)$ $$ \mu^{*}(A) = \left\{ \begin{array}{l ...
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23 views

Is a prior distribution always a random probability measure?

Let $(\mathcal{X}, \mathcal{B})$ be a measurable space and let its probability measure be $P$. In Bayesian statistics, we may wish to define a prior $\mu$ on the space of all such probability ...
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33 views

The mean of $\mu_{P}(\theta)=\frac{1}{Z}P(x|\theta)$

Consider a parametrized probability measure $P(x|\theta)$, that is for each $\theta\in[a,b]$ it is a valid probability measure on $x$. Denote $f(\theta)$ its mean and $\Sigma(\theta)$ its variance. ...
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40 views

Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
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19 views

How to get a combined opinion from 3 different opinions in a reasonable way?

Suppose 3 persons, A, B and C, observe a number and make statement about what they see. The number can only be 0 or 1. What we know about these 3 persons is: If A says 1, the number is 95% possible ...
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33 views

Showing transition function is a transition semigroup

Suppose we have two measurable spaces $(S,\mathcal{S}),\,(T,\mathcal{T})$ and a measurable function $f:S\to T$. Define a probability kernel $\phi$ from $S$ to $T$ by $\phi(x,\cdot) = \delta_{f(x)}$. ...
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13 views

Uniform distribution on spheres in $\mathbb{R}^d$ and Normal random variables/Brownian motion.

Suppose $\lambda(y,\cdot)$ is the uniform distribution on the sphere of radius $y$ in $\mathbb{R}^d$, and $\mu_t(x,\cdot)$ is the distribution of a $N(x,tI)$ random variable in $\mathbb{R}^d$. ...
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23 views

Conditional probability density function of two variables

Assume I am given a continuous function $f(x_1,\cdots x_n)$ over some $n$-dimensional region $R$. I want to show that this function is a jointly continuous probability density function. May I show ...
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28 views

Prove that $\mathcal F_1\otimes\cdot\cdot\cdot\otimes\mathcal F_n=2^\Omega.$

How can I prove that if $n\in\mathbb{N}, i\in\{1,...,n\},$ $\Omega_i=\{0,1\},$ $\mathcal F_i=2^{\Omega_i}=\{\emptyset, \{0\}, \{1\}, \{0,1\}\},$ ...
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26 views

Prove that $C=\left\{x\in L_1:\|x\|_1\le1\right\}$ is not uniformly integrable.

We know that if $p>1$, then $C=\left\{x\in L_p:\|x\|_p\le1\right\}$ is uniformly integrable. How can I prove that if $p=1$, then $C$ is not uniformly integrable. Thanks for your answers.