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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Trouble understanding the theory behind negligible functions and their applications in cryptography

I was formally taught that: $\epsilon$ is a function $\epsilon\colon \mathbb{Z^{\geq0}}\rightarrow \mathbb{R^{\geq0}}$ and if $\exists$d: $\epsilon$ ($\lambda$) $\geq \frac{1}{\lambda^{d}}$ then ...
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110 views

Computation of $\mathbb{E}[\min(U+W,V+W)]$

Our exam today contained a few things I wasn't able to compute starting with this: Let U,V,W,Y,Z be independent random variables with the following distributions $U\sim\text{Exp}(1/5)$ ...
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170 views

Klenke's construction of Brownian motion

Why does Klenke's concise construction of Brownian motion via probability transition kernels satisfy the motion's characterizing properties, equations (14.17) and (14.18)? (results referenced in the ...
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Intuition behind the concept of indicator random variables.

I am studying Randomized Algorithms chapter in the book "Introduction to Algorithms" by Cormen et al. In this chapter the book introduces the concept of an indicator random variable and state that ...
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211 views

Klenke's proof for “Kernel via a consistent family of kernels”

I'm trying to understand a proof in Achim Klenke's textbook Probability Theory: A Comprehensive Course (Springer, 2008). The proof in question is the one for Theorem 14.42, "Kernel via a consistent ...
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147 views

Reference request for examples of probabilistic heuristics, help put some examples in a broader context.

I was thinking about how probability is used in heuristic arguments, an example being the argument that there are an infinite number of twin primes: the probability that $n$ is the first of two twin ...
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700 views

Exponential Distribution Maximum Likelihood

I found the following question in a past exam paper and I would like to ask how to solve it as I can't find anything in the notes related to it: ...
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1answer
842 views

To show that a given process is Gaussian

Suppose I have given a Brownian Motion $W$, this is a Gaussian process, and I define: $$B_s:=W_{t-s}-W_t$$ for $0\le s\le t$. Clearly this random variable has expectation zero. For the covariance ...
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708 views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
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48 views

For any $c_1,c_2\in\mathbb{C}, E(c_1Z+c_2)=c_1E(Z)+c_2$

When dealing with real-valued RVs, the extensions of expectation and variance are quite clear to me. For example, showing $E(aX+b)=aE(X)+b$ and $var(aX+b)=a^2var(X)$ is relatively straightforward to ...
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Characteristic functions of random variables (Poisson, Gamma, etc.)

My self-study in measure and probability theory as finally brought me to the subject of characteristic functions, and I have not handled these in the past with any rigor at all, so all of this is ...
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112 views

Proof: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$

As the title states, the problem at hand is proving the following: $X\ge 0, r>0\Rightarrow E(X^r)=r\int_0^{\infty}x^{r-1}P(X>x)dx$ Attempt/thoughts on a solution I am guessing this is an ...
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1answer
170 views

First hitting time of leftcontinuous process

Suppose that we are working with a Filtration which is right continuous. I know then, that the first hitting time of a right continuous process into an open set is a stopping time. Is the same true, ...
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1answer
99 views

Expectation and proofs on $(\Omega,\mathcal{B},P)$ involving moments and MGF

Ok, suppose we have a random variable, X, on $(\Omega,\mathcal{B},P)$ and $r>0$. I am trying to prove the following 4 things: 1- If $E(|X|^r)<\infty$ then $E(|X|^s)<\infty \;\;\;\forall ...
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722 views

Randomly selecting a natural number

In the answer to these questions: Probability of picking a random natural number, Given two randomly chosen natural numbers, what is the probability that the second is greater than the first? it ...
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1answer
428 views

Are Cumulative Distribution Functions measurable?

It is well-known that CDFs (Cumulative Distribution Functions) of one dimensional random variables are Borel measurable. But does the same apply to CDFs of multi-dimensional random variables (rvecs)? ...
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137 views

Why is ergodicity of transformations only defined for measure-preserving transformations?

In ergodic theory, why does the defintion of an ergodic transformation $T$, why do I have to claim that it is measure-preserving? E.g. $T$ is ergodic if $\mathbb{P}(A) \in \{0,1\}$ for all $A$ ...
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193 views

Property of the sharp bracket process

I know that for a continuous local martinagle $M$ we have $\langle M\rangle^\tau = \langle M^\tau\rangle$ for any stopping times. Now if $M,N$ are two local martingale I know that there exists again ...
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801 views

Supermartingale with constant Expectation is a martingale

In my lecture notes they use the fact, that every supermartingale $(M_t)$ for which the map $t\mapsto E[M_t]$ is constant is already a martingale. Unfortunately I can't prove it. Some help would be ...
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Understanding measures on the space of measures (via examples)

Let $X$ be a Polish space. If it allows for an interesting answer, you may assume $X$ is compact or even $X=[0,1]$. The space $\mathcal{P}(X)$ of Borel probability measures on $X$ is also Polish (via ...
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151 views

Formalizing the shift operator

I hope you can help me formalize some things about the shift operator. So let $(\theta _{n})_{n\geq0}$ be the shift operator - that is $\theta _{n}\omega(k)=\omega(k+n)$. I'm using the Durrett setup ...
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306 views

Show the result of the following infinite sum, based on a binomial random variable conditioned on a Poisson random variable

$$\sum_{n=0}^\infty \binom{n}{k}p^k(1-p)^{n-k}\frac{\lambda^ne^{-\lambda}}{n!} = \frac{(\lambda p)^ke^{-\lambda p}}{k!}$$ That is, given a random variable $X$ with Poisson distribution $X \sim ...
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1answer
889 views

Expected Value of function of two random variable

Assume $X_1$ is an Exponential random variable with unit mean ( i.e. $f_{X_1}(x) = e^{-x}$ ) and $X_2$ is an Erlang distribution with shape $N$ and unit rate ( i.e. $f_{X_2}(x) = ...
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146 views

Sum converging to a Normal

Say $(X_i)$ is a sequence of $n$ independent Bernoulli random variables, with parameters $p_i$, $i:1...n$. How do I prove that the random variables, $$ S_{n} =\frac{1}{n} \sum _{i=1}^{n}a_iX_i $$ ...
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430 views

How to conceptualize conditional expectation inductively?

Attempting the solve the following problem A fair die is successively rolled. Let $X$ and $Y$ denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find $E[ X|Y ...
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Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
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137 views

Expected length of a game of Kings

The game of Kings is a drinking game played with a standard 52-card deck. The rules are irrelevant to the nature of this question; we only wish to calculate the expected length of a game of Kings. The ...
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7k views

conditional and joint probability (3 variables)

I'm having troubles verifying why the following is correct. $$p(x,y|z)= p(x|y,z)p(y|z)$$ I tried grouping the (x,y) together and split by the conditional which gives me ...
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140 views

Convergence of var$(\bar{X}_n)$

$\newcommand{\var}{\operatorname{var}}$ Given $X_1,X_2,\ldots$ is a sequence of random variables with $\rho=0$ and finite second moments. I am trying to show that $$\var(\bar{X}_n)\rightarrow ...
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Demonstrating independence and expectation without Fubini's Theorem

Suppose X and Y are independent random variables, and let f and g be measurable functions, which are either bounded or non-negative. The problem is to show that: $E(f(X)g(Y))=E(f(X))E((gY))$ but I ...
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114 views

Show that $E(|X|)<\infty$ and $E(X_n)\rightarrow E(X)$

After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these ...
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115 views

Proof of convergence of a sum of mean-consistent estimators

After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these ...
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1answer
176 views

When is there in a probability space no null sets?

I remember my lecturer saying that in some cases there will be no other null set than the trivial one (the empty set), but I can't remember exactly the condition. I've been thinking and convinced my ...
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Disintegration theorem, a reference needed

I've come a cross the so called 'disintegration theorem' in multiple occasions now and I'm interested to learn its proof and more on related topics. Particularly I'm interested in a proof for the ...
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3answers
789 views

Finding the expectation of a truncating event given a particular outcome?

Approaching the following problem: Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let $W$ denote the net winnings of a gambler whose ...
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228 views

original source for the Borel-Kolmogorov paradox

Does anyone know the original source for the Borel-Kolmogorov paradox? Is it online somewhere? Kolmogorov doesn't give a precise citation. (He does list three works by Borel in his bibliography, ...
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74 views

Integral of unimodal functions $f>0$.

Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is positive almost everywhere and integrable. We know that if $n=1$ and if $f$ is unimodal then the integral $F(x)=\int_{[-\infty,x]} f$ is convex for ...
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175 views

Inequality regarding difference of characteristic functions

We want to show if $X, Y$ are random variables defined on a common probability space, with characteristic functions $f, g$ respectively, then the following inequality is valid: $$\sup |f(x)-g(x)| \le ...
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285 views

measuring distance between probability measures only at the tail

Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support? Take, for example, the total ...
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268 views

How follows the Strong Law of Large Numbers from Birkhoff's Ergodic Theorem?

We want to prove the strong law of large numbers with Birkhoff's ergodic theorem. Let $X_k$ be an i.i.d. sequence of $\mathcal{L}^1$ random variables. This is a stochastic process with ...
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The Star Trek Problem in Williams's Book

This problem is from the book Probability with martingales by Williams. It's numbered as Exercise 12.3 on page 236. It can be stated as follows: The control system on the starship has gone wonky. All ...
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51 views

What are general conditions for the $L_p$ convergence of an independent random series?

Let $X_1, X_2, \ldots$ be independent $L_p$ random variables. I'm looking for useful conditions which imply $S_n = \sum_{i = 1} ^ n X_i$ converges in $L_p$ to some random variable $S$. If it is ...
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58 views

Continuous variables and proportionality

I have another problem in a statistics past paper that goes as follows: Let $X$ be a continuous random variable, taking values in the range $[0,1]$ with pdf given up to proportionality by $f(x) ...
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814 views

Drawing balls with replacement

I have the following question in an exam paper and I don't know exactly how to treat it. I would appreciate a hint with regards to what formula to use or such: ...
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412 views

Expectation and Median (Jensen's inequality) of Spacial Functions

I hope this forum will be able to help me- if we have a 1-Lipschitz function $f:S^n \to \mathbb{R} $ , when $S^n$ is equipped with the geodesic distance d and with the uniform measure $ \mu $ . ...
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192 views

Detail in definition of stochastic independence for families of events

In a probability theory script, I read a definition of independence where I don't understand one detail (Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space) "A family of events $(A_i)_{i ...
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69 views

Independence through conditional expectation

I have a sequence of random variables $Y_t$ with the same distribution. Is there a way to show that they are independent (i.e., are an iid sequence), if we know that: $\forall i \in \{1,\ldots,n\}: ...
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289 views

strong and weakly orthogonality for martingales

If I have two RCLL martingales $X,Y$, both bounded in $L^2$, hence uniformly integrable. Then we call $X,Y$ weakly orthogonal if $E[X_\infty Y_\infty]=0$ and we call $X,Y$ strongly orthogonal if $XY$ ...
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163 views

Outcome of unmeasurable probability

I consider a standard normal random variable $X$ and a Vitali set $V$. $P(X\in V)$ can not be computed as $V$ is not measurable. Now I consider the outcome of the following experiment $E_S$ : $N_S$ ...
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Moment generating function for the uniform distribution

Attempting to calculate the moment generating function for the uniform distrobution I run into ah non-convergent integral. Building of the definition of the Moment Generating Function $ M(t) = E[ ...