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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Approximation of Conditional Expectation with Respect to “Y” Using Simple Approximation of “Y”

Background. (TL:DR you can skip to Question. below.) This is a followup question to one of my previous questions (linked here) on this website. In short, the other question was about how to express ...
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1answer
10 views

Does this integral $\int f_{X|Y}(x|y) dy$ has any meaning in probability or statistics

Suppose I have two random variables $(X,Y)$ with joint probability density given by $f_{X,Y}(x,y)$. Does integral \begin{align*} \int f_{X|Y}(x|y) dy \end{align*} evaluate to something or has ...
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Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
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1answer
21 views

Assumptions of a probability distribution

Let $X$ be a continuous real-valued random variable indicating the fragility of a firm. Suppose that the firm defaults if $X$ takes a value above a threshold $u>0$. Hence $$ Prob(X>u) $$ is the ...
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1answer
15 views

How is it possible to write $\text {Pr} [M = m]$ where $M$ is random variable defined over a message space $\mathcal M$ and $m \in \mathcal M$.

In cryptography we consider random variables $K, M$ and $C$ over the key space $\mathcal K$ , message space $\mathcal M$ and cipher space $\mathcal C$, respectively. I've studied discrete mathematics ...
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1answer
24 views

Sufficient conditions for monotonicity with probability distributions

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ ...
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21 views

Is conditional Prob less than unconditional prob? [duplicate]

Suppose $X_{n}=1$ with probability $p_{n}$ and zero with probability $1-p_{n}$. Let $F_{n-1}$ be the sigma algebra generated by $X_{1}, X_{2},...,X_{n-1}$. Then is that true $E(X_{n}| F_{n-1} ) ...
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1answer
32 views

Algorithm for risky investments in banks

I made the following programming question on stack overflow but the users said it was more of math question. Here it is. Situation You start with a fixed amount of money, take it as $\$1000$. You ...
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1answer
32 views

probability of randomness [on hold]

If you eat three apples, two squares, and seven artichokes, what is the probability that you will become green before you become seventy. I would like real thoughtful answers. Thanks in advance.
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1answer
16 views

Infinite products of scaled indicator variables: almost sure convergence vs. uniform convergence of the sample mean

Let $\frac{X_i}{2}\sim Ber(0.5) \implies E[X_i]=1$, and let $Y_n=\prod\limits_{i=1}^n X_i$. Since the $X_i$ are iid, $E[Y_n]=1,\;\forall n<\infty$. However, something weird appears to be happening ...
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1answer
24 views

Problem 4.2 (p. 60) in Karatzas and Shreve

I'm looking at problem 4.2 in "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve. The goal is to show that on $C[0,\infty)$, the Borel sigma algebra generated by "topology of local ...
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1answer
29 views

Find expected value of $W$, when $ W $ is the largest of the variables. [on hold]

Let $X_1, X_2,\ldots, X_8$ be independent exponential random variables of mean $1/2$, Let $W$ be the largest of the $X_1, X_2, \ldots, X_8$. Compute the expected value of $W$.
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1answer
20 views

Probability generating function for pascal distribution

The objective is to find the P.G.F of the Pascal($n,p$). $n = 1,2,3\ldots$ $p$ $\in$ [0,1] and $q = 1-p$ $p_x(k)=P(X=k)=\binom{k-1}{n-1}p^nq^{k-n}$ $k = n,n+1,n+2,\ldots$. ...
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0answers
26 views

Stronger version of strong law of large numbers

Let $(X_i)_{i\in\mathbb{N}}$ be pairwise independent random variables where $E[X_i]=0$ for all $i\in\mathbb{N}$ and $\sup_{n}E[X_n^2]\lt\infty$. Then for $S_n=\sum_{i=1}^n X_i$ and ...
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0answers
11 views

Looking for solutions manual: Probability and Stochastic Processes for Engineers [on hold]

My first posting to this community. I am an engineer. I am trying to teach myself the elements of Stochastic Processes. I found the book "Probability and Stochastic Processes for Engineers" By C. ...
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0answers
9 views

Increments of a Brownian motion involving stopping times

I don't quite understand a proof involving Brownian motion in my book: Let $B$ be a standard Brownian motion and let $T$ be an a.s. finite stopping time. For some fixed $n \in \mathbb{N}$, let $T_n = ...
3
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1answer
15 views

Independence of random variables involving Brownian motion

I am reading a book on stochastic analysis and I don't understand the following (i.e. don't know how to prove it rigorously): Let $B$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the ...
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0answers
20 views

We place uniformly at random n points in the unit interval [0, 1]. [on hold]

How to go about the question when it asks: Denote by random variable X the distance between 0 and the first random point on the left. What is the probability distribution function FX(x) and pdf?
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23 views

Express expected value with help generating function

I understand, how to express expected value with help generating function. For example, I have the following generating function: $D(z) = p K(z) + q M(z)$, where $p + q = 1$. How can I express ...
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1answer
47 views

Linear function and expectation

At first we have a function f supposed to be convex. Show that if $E(f(X))=f(E(X))$, where X is a random variable, it implies that $X=E(X)$ almost surely. $E(f(X))=f(E(X))$, by Jensen's inequality, ...
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25 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
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0answers
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Measurability of the points of (strict) increase for Stochastic Process

Given a background space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ , I'm considering a stochastic process $X:=(X_{t})_{t\geq0}$ with distribution $X(\mathbb{P})$ on ...
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2answers
48 views

Probabilistic implications of the existence of non-measurable sets

Measure theory and probability theory are deeply connected through the interpretation of subset measures on the sample space as probabilities of events. A major (and somewhat disturbing) result from ...
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1answer
62 views

Show that $E(X|Y, Z) = E(X|Y)$ almost surely with condition Z is independent of $(X, Y)$

$(X, Y, Z)$ is a continuous random vector and $Z$ is independent of $(X,Y)$. Prove that $E(X|Y, Z) = E(X|Y)$ almost surely. I had been thinking this question tonight but couldn't figure out how to ...
3
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2answers
67 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
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0answers
54 views

Is it normal (correct) to calculate a probability without knowing the sample space?

Is it normal (correct) to calculate a probability without knowing the sample space? Background: I have finished a probability calculation $\mathbb{P}(E)$. I want to do some simulations. ...
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1answer
43 views

Definition of conditional probabiliy as function dependent on $\sigma$-Algebra

I know that for events $A,B$ with $P(B) > 0$ the conditional probability is defined as $$ P(A | B) = \frac{P(A \cap B)}{P(B)}. $$ Of course by regarding $A$ as constant, and varying $B$ we get a ...
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36 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf ...
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limsup $X_n/\sqrt{\log(n)}$ didn't get answers from previous question [duplicate]

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
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Standard machine in measure theory

Step 1.Prove the property for $h$ which is an indicator function. Step 2.Using linearity, extend the property to all simple positive functions. Step 3. Using Monotone property extend the ...
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1answer
20 views

Sum of uniformly distributed random variables in a given range

I am trying to find the sum of n uniformly distributed i.i.d random variables in the range [0-W]. I am aware that if the variables are distributed in the interval (0,1) then their convolution is given ...
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3answers
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Indicator in expectation

Suppose we have a measure space $(\Omega,\mathcal{F},P)$, say we have a random variable $X$ defined on this measure space. My question now is; if we have an event say $F \in \mathcal{F}$ is it in ...
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Functional derivative of $\int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx$ with respect to $f_X(x)$

What is functional derivative of \begin{align*} \int f_{X,Y}(x,y)e^{-f_{Y|X}(y|x)} dx \end{align*} with respect to $f_X(x)$. Here $f_{X,Y}(x,y)$ is joint probability density of r.v. $(Y,X)$ and ...
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2answers
36 views

Suppose $P(X \in B) \in \{0,1\}$ for all $B \in \mathcal B(\mathbb R)$. Show $X = c$, $P$-almost-surely.

Let $(\Omega, \mathcal F, \mathcal P)$ be a probability space and let $X$ be a random variable. Suppose $P(X \in B) \in \{0,1\}$ for all $B \in \mathcal B(\mathbb R)$. I want to show that there ...
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1answer
29 views

How to show the expected value of a hitting time Brownian motion?

We have $W_t$ as a Brownian motion and $$T_{−a,b} = \inf \{t ≥ 0 : W_t \not\in [−a, b]\}\qquad a, b > 0$$ How do you show $\mathbb{E} (W_{T_{-a,b}}) = 0$?
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1answer
63 views

Approximate normal distribution

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
4
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1answer
60 views

notation (ab)use for random variables, distributions, pdfs/pmfs

This question is about notation for random variables (RVs), distributions and pdfs/pmfs and their common (ab)use as I recently got confused. Let $X,Y$ denote random variables. First, notations I ...
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29 views

Supercritical bond percolation on a square lattice in constrained geometry

Consider bond percolation on the square lattice (with, as usual, set of vertices ${(x,y):\; x,y \in Z}$ with parameter $p>1/2$. Let $\alpha>0$ and let A be the event that there is an infinite ...
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1answer
35 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
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A problem on convergence.

Let $X_n$ be an iid sequence of non negative random variables (with $X_n<\infty$ almost surely) which have a common distribution with independent random variable $X$. How to prove the following: ...
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1answer
24 views

Conditional Expectation with Respect to “Y” as a Polynomial in “Y”?

I was reading on conditional expectation online when I came to this curious passage: I can easily understand that $\mathbb E[X|Y]$ can be seen as a function of $Y$: for any $\omega\in\Omega$ in the ...
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1answer
40 views

Almost sure convergence problem.

Let $(X_n)_{n\geq 1}$ be independent random variables: $X_n=n^2-1$ with probability $\frac{1}{n^2}$ and $X_n=-1$ with probability $1-\frac{1}{n^2}$. Let $S_n=\sum_{k=1}^{n}X_k$. How to prove that ...
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0answers
39 views

Central limit theorem extends to absolutely continuous measures

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\mathbb{Q}$ be a probability measure on $(\Omega, \mathcal{F})$ that is absolutely continuous w.r.t. $\mathbb{P}$. Let the ...
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1answer
24 views

How to find transition probability matrix $P$ by using transition rate matrix $T$?

Let $$T = \left(\begin{matrix} -2 & 1 & 1&0 \\ 2 & -3 & 1&0 \\ 1 & 2 & -4 & 1\\ 1 & 3 & 1 & -5\end{matrix} \right) $$ be a transition rate matrix of ...
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16 views

Gibbs Sampler integral computeable

here is an example of a changepoint in a poisson world with the gibbs sampler, it is an bayesian approach. the data are assumed to follow this distributions : $\begin{equation} \nonumber Y_i \sim ...
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1answer
33 views

Joint density invariant under orthogonal transformations [duplicate]

I have a problem and I am totally stuck! I have to show that when the distribution of two random variables $X$ & $Y$ given by $g(x,y)=f(x)f(y)$ is invariant to orthogonal transformations, then it ...
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1answer
24 views

Understanding an application of Entropy

I'm struggling with the following exercise on entropy. Suppose that your friend Alice chooses a number $X$ uniformly at random from $[1,n]$, which she writes down using $\log n$ bits; you can assume ...
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1answer
10 views

Positive component of a submartingale is a submartingale

I am trying to prove the Doob's Upcrossing Lemma and the first step requires to prove that: If $X$ is a submartingale, then $(X-a)_+$ is a submartingale. I found it intuitive but i failed to prove. ...
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0answers
34 views

The efficiency property of estimator for second moment. [on hold]

Please help to improve the efficiency property of estimation for second moment. Statistical population is normally distributed. Sorry for my bad english, if something is wrong. Thanks in advance.
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0answers
10 views

I want Find finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions [on hold]

Write the finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions. Thanks for help.