Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use ...

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How to calculate the probability that the average of a multinomial process exceeds some value

I've been mulling over a problem that has something like the following form. I don't have a math or stats background so advice and answers at various levels, from terminological to strategic, would be ...
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1answer
62 views

Proving a statement about stopping times

I'm getting stuck on the following statement Suppose that $\tau$ is a stopping time on some filtered probability space $(\Omega,\cal{F},\cal{F}_t,\mu)$ and that $f:[0,+\infty]\to[0,+\infty]$ is a ...
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3answers
326 views

Understanding the Gamma function? [duplicate]

I'm working my way through a probability textbook, and i have encountered the Gamma function through the Gamma distribution. I understand that the Gamma function is an interpolating function that ...
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0answers
73 views

A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
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1answer
31 views

Modeling 2 conditionally dependent variables

I am working on image processing and my probability theory knowledge is low. My question here is I am working with 2 variables X and Y which is dependent on each other. That is we can compute P(X|Y) ...
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1answer
52 views

A basic question on base-$r$ non-terminating representation of a number

For $i=0,\dots,r-1$, let $A_r(i_1,\dots,i_k)$ consist of the numbers in the unit interval in whose base-$r$ expansions the digits $i_1,\dots,i_k$ nowhere appear consecutively in that order. I need to ...
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1answer
223 views

Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
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1answer
126 views

Is a compensated Poisson process uniformly integrable

Let $(N_t)_t$ be a Poisson process with intensity $\lambda$. Define $$ \bar{N}_t = N_t - \lambda t $$ which is clearly a martingale. My question is: is $\bar{N}$ uniformly integrable? I strongly ...
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1answer
114 views

Are all small probabilities incomputable?

From Nassim Nicholas Taleb. in Opacity: I spent the last two decades explaining (mostly to finance imbeciles, but also to anyone who would listen to me) why we should not talk about small ...
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1answer
73 views

Which different probabilistic bounds/inequalities apply when we are given a lower bound on the sample size

Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that: $$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$ My question is, what type of probabilistic inequalities require some ...
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2answers
152 views

Bounded logarithmic function

I am trying to find any function that it grows logarithmically up to a certain point, and after that point it remains constant. Can anyone help me with that
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2answers
39 views

Rational probabilities

A probability space is generally defined to be a sample space $\Omega$ with a sigma algebra $F\subseteq 2^{\Omega}$ and a probability function $P:F\to \mathbb{R}$. Is there anything inconsistent ...
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159 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
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1answer
64 views

Regular conditional probability living on sections

Let $X$ and $\bar X$ be standard Borel spaces, let $A\subseteq X\times \bar X$ be an analytic subset of the product space and let $P$ be a probability measure such that $P(A) = 1$. Does there exists a ...
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1answer
125 views

Log-likelihood function

I'm not sure if this could be asked here, or in math overflow... In the following paper Cho, Jin Seo, and Halbert White. "Testing for regime switching." Econometrica 75.6 (2007): 1671-1720. doi: ...
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39 views

Conditional expectation and Gaussian vectors

I have the following problem: Let $X,Y,Z$ be iid Gaussian rv and $U = 2X-Y-Z , V=3X+Y-4Z$ So, I know that (U,V) is a Gaussian vector. The goal is to calculate $E(V|U)$ The hint says to write it as ...
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1answer
58 views

A small question about convergence in distribution

If we have two sequence of random variables $X_n$ and $ Y_n$ such that $P(|X_n - Y_n|) > \epsilon \to 0$ for any $ \epsilon > 0$. If $X_n$ converges in distribution to some distribution (for ...
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4answers
100 views

Boundness & integrability

Let's take a function $f\in L^1$ Does it follows that f is also bounded? Couldn't it be unbounded on zero-sets? I'm working in probability theory (finite measure spaces) Thanks for any help
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1answer
30 views

Inversion formula as a way to check if a function is a characteristic function

Using the notion of characteristic functions from probability theory, suppose that a function from $\mathbb{R}$ to $\mathbb{C}$ is given. Lukacs says, in his book on ch. f., that the usual inversion ...
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1answer
174 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
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1answer
46 views

Pre-image of conditional expectations

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, $S:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable map. Let $\mathcal F_0\subseteq\mathcal F$ and $H\subseteq\mathbb R^d$ a Borel ...
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2answers
332 views

Local martingale is locally uniformly integrable martingale?

Is a local martingale locally uniformly integrable martingale ? Here I define a local martingale to be the process with a localizing sequence $\tau_n$ such that the stopped process is martingale. ...
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2answers
45 views

Random variable independent of its own sign.

Let $X$ be any random variable and assume $X$ is independent of $\text{sgn}(X)$. What is known about $X$ in this case? I'm guessing its equivalent to symmetry, but I haven't had luck proving it.
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1answer
33 views

Estimator of a Random Variable

Given a random varable $Y$ where $$ f_Y(y) = \begin{cases}e^{-(y-k)} \quad x>k\\0\quad \text{otherwise}\end{cases} $$ Given $n$ observations of $Y$. Is the sample mean $\bar{Y}$ an unbiased ...
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1answer
247 views

Equivalent defining Markov property

Consider the stochastic process $(X_t)_{t \in \mathbb{R}}$ and show the equivalence of the following two Markov properties: (a) $P(X_t \in A \mid X_u, u \leq s) = P(X_t \in A\mid X_s) \qquad ...
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2answers
163 views

Given four independent random variables $X_1,\cdots,X_4$ find $Y=\min\{X_1,X_2,X_3,X_4\}$

Suppose that we're given four independent random variables $X_1,X_2,X_3$ and $X_4$ and their probability density function is given by: $f(x)= 3(1-x)^2 $ for $0<x<1$ and otherwise $f(x)=0$. If ...
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1answer
37 views

On the structure of some subsets of the unit interval specified by binary expansion

How to prove that $\{\omega : |\frac{1}{n}\sum_{i=1}^{n}d_i(\omega)-\frac{1}{2}|\geq \epsilon_n\}$ consist of finitely disjoint union of intervals (where $\epsilon_n$ is a monotonically decreasing ...
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2answers
79 views

Probability Qual problem: $X_n\rightarrow X$ in probability implies $EX_n\rightarrow EX$ under given condition?

Suppose $X_n\rightarrow X$ in probability, and there is a continuous function $f$ with $f(x)>0$ for large x with $\dfrac{\vert x\vert}{f(x)}\rightarrow 0$ as $|x|\rightarrow\infty$ so that ...
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1answer
130 views

Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...
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1answer
138 views

Source for Probability of one point in Distribution is zero

I know from my old studies that the probability of one single point, for instance in a normal distribution and quasiprobability distribution, is zero. Where should you cite for this fact? I have ...
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1answer
71 views

How can I prove this equivalence concerning ergodicity?

I write you, because I have a problem to show two equivalences. But before writing them down, I give you all the definitions we had as background: (I) The Quadruple $(\Omega,\mathcal{A},\mu,T)$ is ...
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1answer
201 views

Proof of uniqueness of the extension in Kolmogorov extension theorem

Statement of the theorem. The proof is mainly focused on showing that the candidate probability measure defined on the algebra of sets is $\sigma $-additive. At the end, the Hahn-Kolmogorov theorem ...
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145 views

Conditional Probability Summation Rule Problem

A crime is committed by one of two suspects, A and B. Initially, there is equal evidence against both of them. In further investigation at the crime scene, it is found that the guilty party had a ...
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1answer
142 views

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,...,$X_m$ be i.i.d. F, $Y_1$,...,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z'_1$ + $\Delta$, where $\psi$ is an unknown ...
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1answer
165 views

If $X$ and $Y$ are independent. How about $X^2$ and $Y$? And how about $f(X)$ and $g(Y)$? [duplicate]

If $X$ and $Y$ are independent. How about $X^2$ and $Y$? And how about $f(X)$ and $g(Y)$? I always have confusion about it. I feel ... yeah of course $f(X)$ and $g(Y)$ are independent, because $X$ and ...
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2answers
43 views

On weak convergence of probability measure

Given probability density function(pdf) $f_n,g:\mathbb{R}\to\mathbb{R}$, if there exists some constanct $c>0$ such that $f_n \to c\cdot g$ pointwise, do we necessarily have $c=1$? Applying Fatou's ...
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1answer
90 views

Help proving that a measure is absolutely continuous with respect with respect to another measure

Suppose that $E$ is a locally compact and separable metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In ...
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2answers
43 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
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1answer
257 views

a.s. convergence of sum of normal random variables

From Resnick's A Probability Path, Exercise 7.7.14: Suppose $\{X_n, n \ge 1\}$ are independent, normally distributed with $E(X_n) = \mu_n$ and Var$(X_n)=\sigma^2_n$. Show that $\sum_n X_n$ ...
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1answer
73 views

Reference Request: Semi-Rings and Rings (System of Sets, not Algebraic Structures)

I studied Probability Theory (from a Measure Theory viewpoint) using only Sigma-Algebras. Recently, I got a book about measure theory that starts from Semi-Rings, but it's presentation is too ...
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1answer
42 views

Empirical distribution. Problem with changing variables

We have iid random variables $X_1, X_2, \ldots, X_n$ with continuous cdf $F$. Define empirical distribution function $\hat{F}_n (x)= \frac{1}{n} \sum_{k=1}^n \mathbb{I}_{\{X_i \le x \}}$. Let's ...
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1answer
42 views

Independence of complex random variables

How is the independence defined for two complex valued random variables $Z_1=X_1+iY_1$, $Z_2=X_2+iY_2$? Do we have $E[Z_1Z_2]=E[Z_1]E[Z_2]$?
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1answer
36 views

finding probability mass function

I need help with part (i) I can see that the number of head form geometric distribution. Let Y be the number of heads. Then $P(Y=k) = (1-p)^{k-1} p $ . So the winner will get $10k$ pounds. Then ...
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1answer
102 views

Distribution of a lifetime of a system

I have a quick question which I can't figure out how to start. I actually do not understand how to model the probabilities. can anyone help? Thanks. Here it is: A system will function as long as at ...
2
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1answer
126 views

$\mathbb P $- convergence implies $L^2$-convergence for gaussian sequences

Consider $(X_n)_{n \in \mathbb N}$ a sequence of gaussian random variables whose limit in probability exists and is given by $X$. I was interested in showing that in this particular case we have ...
2
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1answer
81 views

A basic question on probability theory

I see in the first chapter of Billingsley's book that the mathematical properties of the sequence of digits in the binary expansion (non-terminating) of $\omega$ where $\omega \in [0,1]$ mirror the ...
3
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1answer
77 views

Mathematics of a Simple Counting Game

I wonder how can one think mathematically about the following game: People sit in a circle. One of them says "One!". Then somebody (no matter who - he/she can even be the former person) says ...
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3answers
131 views

How can I linearly decrease chance?

Suppose there are 10 different people, each holding 1 lottery ticket, and I give each of them 10% chance of becoming the 'winner'. Now let's say one of those 10 people (I dont know which one) ...
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1answer
168 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
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1answer
27 views

Small question about convergence of probalility

In the lecture our professor mentioned that: If $X_1 , X_2, ...$ are iid random variables. Then $ n P (|X_1| \ge a_n) \to 0$ implies $ P(\max _{k \le n} |X_k| > a_n) \to 0$, where $\{a_n\}$ is just ...