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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Basic question about the probability and expectation of a bijective function.

Dear stackexchange community, I am still unskilled in the language of mathematics, in fact probability theory to be precise. In my spare time I like to do some research of my own, and I am having ...
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1answer
37 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
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1answer
24 views

What are the restrictions on infinite sums of measures?

Select a positive natural number at random. For all n, the probability of selecting that number n is zero. Yet the sum for all n of (probability of selecting number n) is 1. So there is some ...
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2answers
58 views

Properties of a sequence of iid rv's

I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
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24 views

Intuitively understanding a probability-problem

Two teams take part at a KO-tournament with n rounds. Assuming, that the teams win all their games until they are paired together, what is the probability that they both meet in the final ? I ...
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18 views

Density of the $k^{th}$ smallest of $X_1,X_2,…,X_n$

Show that if $(X_1,X_2,...,X_n)$ are i.i.d. with common density $f$ and distribution function $F$, then $X_{(k)}$ has density $$f_{(k)}=k\binom{n}{k}f(y)(1-F(y))^{n-k}F(y)^{k-1}$$ where ...
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1answer
22 views

Evaluating an expectation of the supremum of collection of random variables

I know that $\mathbb{E}(sup_n|X_n|)=\infty$ if $X_n=\frac{2^n}n\cdot\mathbf 1_{(1/2^{n+1},1/2^n)}$. However I am not sure how this can be evaluated explicitly. The probability space is $Ω=[0,1]$ ...
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1answer
23 views

Calculation of conditional variance

I'd like to ask, whether I did this task correctly. We have two r.v. $X,Y$. Firstly, $Y$ has Bernoulli distribution $\mathbb{P} \left( Y=1\right)=a \; \; \mathbb{P} \left( Y=2\right)=1-a$ Moreover ...
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1answer
28 views

Intuition of Markov structure and how variables relate in the structure

Assume three r.v. $X_1, X_2, X_3$. They are conditionally independent in the following way: $ X_1 \perp X_3 | X_2$ We have that: $$P(X_3 | X_2) = P(X_3| X_2, X_1)$$ In the notes I am reading it ...
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2answers
31 views

Math probability combination explanation

A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. once she locates the two defectives, she stops ...
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1answer
46 views

First hitting time expectation and Markov property

Let $H_A$ be the first hitting time, such that $H_A\geqslant1$, so we have $X_0=i\notin A$. All texts I looked at, state without any further justification that $$ \mathbb E(H_A\mid X_1=j, ...
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31 views

Convergence of probability density functions

Assume that a sequence of random variables, $(X_t)_{t\geq 0}$, converges in distribution to a random variable $X_0$, as $t\to 0$. Also assume that $X_t$ and $X_0$ have $C^{\infty}$-probability density ...
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36 views

References for Last exit density function for a Borel set A [closed]

I want to find the probability $P_{x}(B(\gamma)\in A,0<\gamma<T))$, where $\gamma=\sup\{t\in [0,T]: B_{t}\in K\}$ be the last exit time of $d$-dimensional Brownian motion $\{B_{t}:0<t<T$ ...
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1answer
43 views

binomial coefficient: maximum value

For $n\rightarrow \infty$ we consider $$f(p)=\sum_{j=c}^n {n\choose j} p^j (1-p)^{n-j}.$$ We are interested in $\hat{p}:=\arg \max_p f(p)$. Can we say something about $\hat{p}$ dependent on $n$ and ...
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1answer
30 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
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1answer
23 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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81 views

Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ ...
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1answer
72 views

Is this true: probability independent from i?

We have a set of i.i.d. random variable $X_i$ with some discrete distribution. Further we have a random variable Y, Independent from $X_i$ with a Binomial Distribution Bin(n,p). Now we are ...
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0answers
23 views

Computing equilibrium measure for Borel sets eg. Ball

I am asking for methods to compute equilibrium measures. The more the better. Here is the definition of equilibrium measure in the Brownian motion setting: Let $\gamma=\sup\{t\in [0,T]: B_{t}\in ...
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35 views

Sums of Power Law random variables

Suppose $F$ be a pareto distribution with scale parameter $x_m$ and shape parameter $\alpha$. Assume $X_1, X_2 , ..., X_n$ are iid random variables drawn from $F$. Let $S_n(k) = X_1 ^k + X_2 ^k + ...
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33 views

A simple (?) equality of two initial $\sigma$-algebras

I'm afraid I miss the forest for the trees.... Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $(E,\mathcal{B}_E)$ be a state space and $\mathcal{E}$ a generating $\pi$-system of ...
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46 views

What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
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13 views

A question in a textbook about Blumenthal 0-1 law for a general Markov process

This question came up as a result of reading this question . Here is the Blumenthal 0-1 law in the book Stochastic Processes by Richard F. Bass. Proposition 20.8 Let $(X_t , \Bbb{P}^x)$ be a ...
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63 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
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1answer
57 views

Precise definition of the support of a random variable

I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \mathcal{F}, Pr)$, where $\Omega$ is the set of outcomes ...
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20 views

Multiple events happening almost surely

I am reading an article about reinforced random walks from B.Davis. In the summary the author basically says that there are two events , I call it $A$ and $B$, and that either $A$ happens almost ...
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1answer
29 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
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1answer
24 views

$L_p$ spaces and tail estimates

I can prove the main identity in this question. Not sure how the "and deduce" bit works. I think $O(\lambda^{-q})$ is some kind of tail estimate.
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1answer
24 views

Probability “average” understanding

This is more of a problem understanding probabilities than an actual question. In a game I am playing I can use a certain item to try to unlock different levels. The item will unlock a new level ...
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63 views

Show independence of $(aX,bX^2)$, $\,X\sim N(0,\sigma^2)$?

How can we proof that $aX,bX^2$ are independent iff $b\cdot a=0$, when $\,X\sim N(0,\sigma^2)$? I found that $X^2$ is Chi-Square distributed, and the correlation is: ...
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1answer
21 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
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5 views

Matrix Completion feature vectors

In the noisy matrix completion problem, with enough number of revealed entries say $|E| = nr^2 log(n) $, can we have a bound on the error in the singular vectors of a sub matrix. For, example say ...
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2answers
34 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
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37 views

Why is the Stopping Theorem interesting?

The theorem for discrete-time martingales is as follows: Let $X=(\Omega,\mathcal{F},(\mathcal{F}_n)_n,(X_n)_n,\mathrm{P})$ be a supermartingale and $\tau_1,\tau_2$ two a.s. bounded stopping times on ...
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1answer
54 views

Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$

Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem? $$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$ with $$ ...
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39 views

Why does a Gaussian process have a gradient whose determinant is Gaussian?

I'm trying to understand something in Adler and Taylor's book, Random Fields and Geometry. Let $T \subset \mathbb{R}^N$ be a compact parameter set (for simplicity, suppose it is a closed hypercube) ...
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20 views

Inverse function of borel sets when function is a constant.

Following a simple proof my professor explained in class I am having problems with a specific step: The proof is of probabilistic nature and we are trying to prove that If $X$ (random variable) is ...
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31 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
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1answer
42 views

Measures in conditional expectation.

I always make confusion when a measure has to be changed in some other measure. This time I'm stuck on a change of measure in the definition of conditional expectation of a random variable. If $Z$ is ...
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32 views

characteristic function of logarithm of random variable

If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$? I know that $\phi_X(t)=E[e^{itX}]$ and ...
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18 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
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68 views

joint probability equals to marginal probability

Is there any situation in that joint probability $p(x,y)$ equals to marginal probability $p(x)$? what is the interpretation of this situation?
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36 views

Square root of Chi-square distribution tends to $N(0,1)$

The question requires to show that $\sqrt{2\chi^2_n}-\sqrt{2n}$ converges in distribution to $N(0,1)$ as $n \rightarrow \infty$, which I dont know how to proceed. The question also has a first part ...
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1answer
32 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
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31 views

An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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1answer
25 views

Characteristic function respect to a product measure.

In my lecture notes of probability theory there is a passage I don't understand: We defined the characteristic function respect to a law of probability of a m-dimensional random variable: ...
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29 views

If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

I was suddenly suspicious of what title says. My logics are follows. $\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s. $\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$ $\Leftrightarrow$ $\exists ...
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Searching for theorems that prove almost sure convergence from convergence in probability

As we can see that almost sure convergence implies convergence in probability, and the converse is not necessarily true. But now I would like to prove a particular sequence of random variable ...
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1answer
66 views

Proving it's a martingale and more conditions.

Let $(X_{n})_{n>0}$ be a sequence of random variables in $[0, 1]$ and assuming that ($X_{0}=a) \epsilon [0, 1]$ then: $Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$ and ...
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23 views

continuous, non-additive set function

I am looking for a non-additive, continuous set function from a simplex $\Pi_{n}$ of finite dimension $n-1$ into $[0,\infty]$. The motivation is as follows. Shannon defined the entropy ...