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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complex integration over the whole plane

I am trying to solve this integral: $H(z)=\int_{\mathbb{C}}{p(z)\log_2{p(z)}dz}$, where $z$ is a complex number with complex normal distribution $p(z)$, and $\mathbb{C}$ denoted the complex plain. ...
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347 views

probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two. Let $X$, $Y$ be two independent random variables with the following characteristic functions: ...
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1answer
128 views

Continuity of Expected Value

Let $m(\cdot)$ be a probability measure on $Z$, so that $\int_Z m(dz) = 1$. Consider a continuous function $f: X \times Y \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subseteq \mathbb{R}^n$, ...
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439 views

recursive equation for number of white balls

Consider a polyurn scheme of more than two colors. Let us draw a ball from the urn and replace it with another ball of the color we picked from the urn. We assume that $w$ is the number of white balls ...
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When can a measurable mapping be factorized?

Problem 13.3 of Probability and Measure by Billingsley states: $(\Omega, \mathcal{F})$ and $(\Omega', \mathcal{F}')$ are two measurable spaces. Suppose that $f: \Omega \rightarrow ...
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$P(\limsup A_n)=1 $ if $\forall A \in \mathfrak{F}$ s.t. $\sum_{n=1}^{\infty} P(A \cap A_n) = \infty$

Let $\mathfrak{F} = (A_n)_{n \in \mathbb{N}}$. Prove that $P(\limsup A_n)=1$ if $\forall A \in \mathfrak{F}$ s.t. $P(A) > 0, \sum_{n=1}^{\infty} P(A \cap A_n) = \infty$. (Side question 1 Is second ...
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1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
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2answers
66 views

Probability Theory - Fair dice

Three fair six-sided dice are thrown and the dice show three different numbers. Find the probability that at least one six is obtained. Im unsure ofwhat type of question this is, I have tried ...
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43 views

chose uniformly at random from the n different brands, independently of previous orders [duplicate]

Michiel's Craft Beer Company (MCBC) sells $n$ different brands of India Pale Ale (IPA). When you place an order, MCBC sends you one bottle of IPA, chosen uniformly at random from the $n$ different ...
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1answer
42 views

A basic question on expectation of distribution composed random variables

Suppose that $X$ and $Y$ are random variables with distribution functions $F$ and $G$. If $F$ and $G$ have no common jumps then I need to show that $E[F(Y)] + E[G(X)] = 1$. How to proceed here ? ...
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2answers
32 views

PDF of the addition of several outcomes from Poisson distribution

We draw $n$ values from a Poisson distribution and add them. - What is the expected of this addition - What is the PDF of this addition It seems quite intuitive to me that if we add $n$ Poisson ...
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1answer
66 views

Expected Value and Variance of Two Random Variable

Let $X_1, X_2,...,X_n$ and $Y_1,Y_2,...,Y_m$ be independent exponential distributed random samples with mean $\theta$. Let $T\alpha = \alpha\bar{x} + (1-\alpha)\bar{y}$, where $0 < \alpha < 1$. ...
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54 views

Transformation of Multiple Variable

I'm having trouble with the following exercise: Let $Y_1, Y_2, \dots, Y_n \overset{\rm i.i.d.}\sim \exp(\theta)$ are random samples. If $Y_i$'s are sorted in ascending order, the ordered random ...
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1answer
61 views

Approximate Expectation of $x^2$

I am estimating the $E[x^2]$. The function I am looking at gives you a constant value of A from 0 to 4, it is 0 from 4 to 6, and A from 6 to 10. I got that $E[x]$ to be 5. I calculated the value of ...
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1answer
61 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
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1answer
242 views

Expected value of c.d.f when normal distributed

I need help to calculate the expected value of an invertal of a c.d.f function which is normal distributed. I know that $E(X)=\int^\infty_0 (1-F(x))dx$ What i need is to calculate $E(w|w \geq ...
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661 views

What is the PDF of a product of a continuous random variable and a discrete random variable?

Let $N$ be a discrete random variable which takes values in [0, ..., M], M > 0, with known PDF $P(N=n)$. Let also the continuous random variable $Z = \sum_{i=1}^{N}X_{i}$ as the sum of i.i.d. $X_{i}$ ...
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107 views

How to model mutual independence in Bayesian Networks?

It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations. Can mutual ...
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210 views

Probability of two opposite events

Suppose there is string of eight bits, e.g.: 00100110 Bits are randomly chosen from the string. All choices are done equally likely. Probability of choosing $0$: $p_0 = \frac{5}{8} = 0.625$ ...
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808 views

pdf equation for tossing 2 coins given the probability of landing head for each coin in a single toss

Problem: Consider a simple coin-flipping experiment in which we are given a pair of coins A and B of unknown biases, $\theta_{A}$ and $\theta_{B}$ respectively (that is, on any given flip, coin A ...
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519 views

Minkowski's Inequality For Infinity

I've tried figuring this out and searching the net on this for 5 hours, but I can't get it. Every source says it's trivial, but I must be missing something because I have pages of work that don't lead ...
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Finding an expression for the probability that one random variable is less than another, given a condition.

Let $X$ and $Y$ be two independent random variables, who's supports are $[0,\infty]$. We can express $\mathbb{P}[X<Y]$ as: $$\mathbb{P}[X < Y] = ...
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Why is this coin-flipping probability problem unsolved?

You play a game flipping a fair coin. You may stop after any trial, at which point you are paid in dollars the percentage of heads flipped. So if on the first trial you flip a head, you should stop ...
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Random Variable Inequality

Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result: (A-S ...
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How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?

I'm learning Kolmogorov's zero-one law in probability theory: Let $(Ω,{\mathcal F},P)$ be a probability space and let $F_n$ be a sequence of mutually independent $\sigma$-algebras contained in ...
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1answer
879 views

What are some open research problems in Stochastic Processes?

I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes. Any examples or recent papers or similar would be appreciated. The motivation for ...
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600 views

On the set of the sub-sums of a given series

Choose a sequence $(x_n)_{n\in\mathbb N}$ of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every ...
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1answer
149 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
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1answer
288 views

Definition of the Brownian motion

The way I understood the definition of a Brownian motion $B_t$ in $\mathbb R$ is that it consists of two parts: We first define the finite-dimensional distributions $$ ...
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132 views

Is there one-tailed version of Vysochanskiï–Petunin inequality, like Chebyshev?

The Vysochanskiï–Petunin inequality gives a tighter bound than Chebyshev for unimodal distributions . I'm just wondering if there is a one tailed version of it, like that of Chebyshev inequality? ...
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167 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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628 views

Cover time chess board (king)

Consider a random walk of a king on a standard chess board, which at each step moves to a uniformly random permitted square. What's the exact mean time to visit all squares (cover time), starting ...
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802 views

Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...
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The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
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352 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
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4answers
2k views

Questions on atoms of a measure

In Kai Lai Chung's A course in probability theory, An atom of any probability measure $\mu$ on $(\mathbb{R}, \mathcal{B})$ is a singleton $\{x\}$ such that $\mu({x}) > 0$. In Wikipedia: ...
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187 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
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1k views

Best fit ellipsoid

Given a collection of points $P \subset \mathbb R^3$, a crude characterization of the "shape" of $P$ is sometimes given by the principal components. We construct a covariance matrix, e.g., if $P$ is ...
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765 views

What distinguishes the Measure Theory and Probability Theory?

It is clear that the Theory of Probability works primarily with limited measures on measurable spaces. On the other hand there is a folklore that says that what distinguishes the Theory of ...
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276 views

Application of Central Limit Theorem

In the book Probability Essentials, by Jacod and Protter, the following question has bugged me for a long while and I'm wondering if it is bugged. The question is an application of Central Limit ...
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1answer
847 views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...
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777 views

Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law

I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law? The former is about limsup of a sequence of events, while the latter is about ...
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191 views

What does the -log[P(X)] mean in the calculation of entropy?

The entropy (self information) of a discrete random variable X is calculated as: $$ H(x)=E(-log[P(X)]) $$ What does the -log[P(X)] mean? It seems to be something like ""the self information of each ...
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Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
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1answer
274 views

Dominated convergence and $\sigma$-finiteness

I am curious about the Dominated Convergence Theorem for a sequence of functions that converges in measure. Theorem: Let $(X,\mathcal{S},\mu)$ be a measure space. If $\{f_n\}, f$ are measurable, ...
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1answer
273 views

How to determine the expected value of the $f(x,y)$?

How to determine the expected value of the $f(x,y)$ defined as: f(x,y): $\quad$ for i = 1 to y $\quad$ $\quad$ do x = R(x) $\quad$ $\quad$ return x where $R(N)$ returns any ...
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4answers
293 views

Expected number of frog jumps

There is frog jumping forward on a line. Each jump distance is random with a known cumulative distribution function $F$. What is the expected number of jumps to reach (or go beyond) distance $d$ from ...
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1answer
268 views

Solution to the stochastic differential equation

Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
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Comparing the expected stopping times of two stochastically ordered random processes (Prove or give a counterexample for the two claims)

Information: a-) $X$ and $Y$ are two continuous random variables on $\mathbb{R}$ having continuous distribution functions $F$ and $G$ with $G(y)\geq F(y)$ for all $y$. b-) $S^X_n=\sum_{i=1}^n X_i$, ...
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105 views

Question about Feller's book on the Central Limit Theorem

My question concerns the proof of Theorem 1, section VIII.4, in Vol II of Feller's book 'An Introduction to Probability Theory and its Applications'. Theorem 1 proves the Central Limit Theorem in the ...