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 [tag:probability-...

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0
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0answers
11 views

probability measures vs. probability distributions vs. measure of probability density

I am learning probability theory right now and am confused about some basic concepts. I have a few questions and am wondering if you can also check if the following is correct: Suppose we have a ...
0
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1answer
19 views

Weak convergence implies convergence on continuous functions

Let $X$ be a metric space, and let $\mu_n$ be a sequence of measures on $X$ converging weakly to a measure $\mu$, meaning for all bounded continuous functions $f$, we have $\int_{X}fd\mu_n \rightarrow ...
1
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0answers
14 views

Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
0
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2answers
16 views

Independence of Random Variables From Expectation Counter Example

I know that if $X$ and $Y$ are independent random variables, then $\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$. I also know that the converse is not true, although I cannot seem to find an easy ...
-1
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0answers
14 views

Communicating classes of a power of the irreducible transition matrix? [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P^k$. In terms of $d$ and $k$, how many communicating classes does $P^k$ have, and what is the period ...
0
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0answers
16 views

Convergence in law and sequence of reals

How to show that if I have $Y_n$ a sequence of random variables and $a_n$ a sequence of reals such that (i) $Y_n\to Y$ in law (ii) $a_n\to a$ i have $a_nY_n\to aY$ in law? I know that if I have (i)...
0
votes
1answer
14 views

Sum of Random Variables i.i.d. with $E[|X_n|]=inf$

Let $(X_n)$ be a sequence of IID RVs (independent, identically distributed random variables) with $E[|X_n|]=+\infty,\forall n$. Prove that $\sum_n P[|X_n|>kn]=\infty$ with $k\in \mathbb{N}$
1
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1answer
26 views

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$?

Does there exist a random variable $\xi$ and a constant $c \neq 0$ such that $\xi + c \stackrel{d}{=} \xi$? For context, I'm re-reading Kallenberg and in Chapter 3, on page 49, in his proof of Lemma ...
0
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1answer
5 views

Indipendent events from any other

In $(\Omega,\mathcal{F},P$) probability space, how can I show that $\forall A\in \mathcal{N}=\left\{ A\in\mathcal{F}: P(A)=0,or\,\, P(A)=1 \right\}\Rightarrow$ $\forall E\in\Omega$ I have $P(A\cap ...
-1
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0answers
32 views

Possible master thesis [on hold]

I am searching a topic for my master thesis. My interests are especially probability theory (something with brownian motion would be nice) and fourier analysis (also in an abstract Hilbert space ...
3
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0answers
19 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ holds $\min_{s\in [0,t]}P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to work, and the ...
0
votes
1answer
35 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu(A\cap T^{-n}B)=\mu(A)\mu(B)$ for all $n\geq N$ for ...
1
vote
1answer
31 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
-1
votes
0answers
29 views

Entropy and Markov chain [on hold]

Assume that $X_n$ is a discrete Markov chain and $H$ is entropy function. I want to prove $$H\left(X_0\mid X_n\right) \geq H\left(X_0\mid X_{n-1}\right)$$ but I have no idea how to prove it. please ...
0
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0answers
22 views

Probability of $n$ numbers picked from set to have greater mean than set

If we have (N) varying non-negative numbers , with a mean equal to X, and a median less than X, if we pick (n) unique numbers from the set, what is the formula for the probability that the mean of ...
1
vote
1answer
48 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
-2
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0answers
17 views

Let U, V and W be i.i.d. random variables with uniform distribution on [0,1]. Find the distribution of $(U V )^W $. [on hold]

Let U, V and W be i.i.d. random variables with uniform distribution on [0,1]. Find the distribution of $(U V )^W$ .
0
votes
1answer
27 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [on hold]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
1
vote
1answer
32 views

$X_1, X_2, \dots$ uncorrelated, $\frac{Var[X_i]}{i} \rightarrow 0$, then $\frac{S_n}{n} - \frac{\mathbb{E}[S_n]}{n} \rightarrow 0$ in $L^2$

Let $X_1, X_2, \dots$ be uncorrelated random variables with $\mathbb{E}[X_i]= \mu_i$ and $\frac{Var[X_i]}{i} \rightarrow 0$, when $i \rightarrow +\infty$. Show that $\frac{S_n}{n} - \frac{\mathbb{E}[...
1
vote
1answer
25 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
0
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0answers
15 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
-3
votes
0answers
21 views

Converges in distribution and probability for constant random variable. [on hold]

Let $X$ be a constant random variable. Then $X_n$ converges in probability to $X$ if and only if $X_n$ converges in distribution to $X$.
0
votes
0answers
42 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
-1
votes
1answer
42 views

In the answer to the question attached below, I don't quite see how step-3 is derived from step-2, Can anyone explain [duplicate]

Calculating the expected values of the min/max of 2 random variables Consider two fair $k$-sided dice with the numbers 1 through $k$ on their faces, obtaining values $X_1$ and $X_2$. What is $\mathbb{...
3
votes
0answers
19 views

Doob Meyer decomposition in an exercize

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
0
votes
0answers
32 views

Function of random variable: Two ways to find the pdf

Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function. Method I first using the ...
1
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1answer
42 views

Complicated probability question [on hold]

There are 20 empty present boxes numbered from 1 to 20 are placed on a shelf, there are 4 men standing in front of the shelf. Each one asked to pick in his mind 3 numbers without telling any one then ...
0
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1answer
34 views

Condition expectation of functions: $E(fg\mid\mathcal{A})=gE(f\mid\mathcal{A})$ when $|g|<\infty$ a.e.

Let $(X,\mathcal{B},\mu)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$\sigma$-algebra, then by an easy application of the Radon-Nikodym Theorem, letting $\nu(A)=\int_A\, f\,\mathrm{...
0
votes
1answer
31 views

Integration with respect to a Poisson random measure

Let $N$ be a Poisson random measure (PRM) on a Polish space, $\left(X,\mathcal{B}(X)\right)$, and let $\tilde{\nu}$ be its mean measure. Then, let $f$ be any non negative and bounded function on $X$. ...
1
vote
1answer
38 views

Why is the CLT stated like it is?

The CLT says that given finite variance of iid RVs, we have $$\sqrt{n}( \bar{X} - \mu) \rightarrow \mathcal{N}(0,\sigma^2),$$ but if this is true, then $\bar{X} - \mu$ should converge to $\mathcal{N}(...
0
votes
1answer
24 views

For an invertible measure preserving system, $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$

For an invertible measure preserving system, show that $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$. Here we consider the measure preserving system $(X,\mathcal A,\mu,T)$ where $T$ is invertible and $\mu$...
1
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0answers
32 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
-2
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0answers
17 views

Probability calculation [on hold]

There are n number of data. n data contain x ideal data and y raw data. Question: how to select randomly one ideal data. Please give me calculation with explanation. Thanks in Advance
3
votes
1answer
46 views

Law of large numbers for moving mean

Consider the following process: For $n = 1,\ldots$ $U_n \sim U[0, 1]$, that is, uniformly distributed on $[0, 1]$, $X_n = U_n 1_{U_n > q_n}$, where $q_n = \frac{1}{n-1} \sum_{i=1}^{n-1} X_i$, ...
5
votes
3answers
72 views

Can $Y$ and $\frac{X}{Y}$ be uncorrelated if neither $X$ or $Y$ is constant?

Suppose I have two variables $X$ and $Y$ with $Y>0$. Can the random variables $Y$ and $\frac{X}{Y}$ ever be uncorrelated, i.e., $$\mathbb{E}(X)=\mathbb{E}(Y)\mathbb{E}\left(\frac{X}{Y}\right).$$ ...
0
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1answer
20 views

Question regarding probability density function?

I was going through the concept of probability density functions and had a small confusion about the notation that a pdf can take values greater than one.I found this How can a probability density be ...
2
votes
1answer
40 views

Is this proof of convergence in probability correct?

${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad $ in probability show that $$\lim_{n\to \infty} \min_{...
4
votes
2answers
33 views

Distribution of throws of die rigged to never produce twice in a row the same result

A die is “fixed” so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probablity 1/5. If the first score is 6, what is the probability that the ...
3
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0answers
34 views

Comparing different definitions of tightness for measures

Let $X$ be a Hausdorff space, $\mathcal{B}(X)$ the Borel $\sigma$-algebra and $\mu : \mathcal{B}(X) \to [0, \infty]$ a measure. Consider the following properties: (1) $\forall A \in \mathcal{B}(X): \...
1
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1answer
23 views

Show that MLE estimator convergences in probability to actual parameter

For iid stochastic variables $X_1, ..., X_n$ whose distribution is defined by 2 parameters, I have found the MLE estimators. They are $\hat{\mu} = \sum x_i/n$, and $\hat{\lambda}$ given by $$ \frac{...
2
votes
1answer
20 views

Probability: Find Dispersion of X + Y

$X = \operatorname{Bi}(3,\frac14), Y=\operatorname{Bi}(4,\frac12), \operatorname{Cov}(X,Y) = -\frac34.$ Dispersion of $X+Y =?$ $D(X) = npq = \frac9{16}. D(Y) = npq = 1$ $D(X + Y) = D(X) + D(Y) = \...
0
votes
1answer
71 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
2
votes
1answer
35 views

An ancillary result from convergence in probability

I was reading a paper concerning probability theory. We have that $X_i$, $i = 1,2,...$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $$\frac{S_n}{n} \...
2
votes
5answers
87 views

Compare $\mathbb{E}[XY]\mathbb{E}[XY]$ with $\mathbb{E}[X]\mathbb{E}[XY^2]$

$\newcommand{\E}{\mathbb{E}}$So this was a question asked to me in an interview where $X$ and $Y$ are two random variables and I was asked to compare the $\E[XY]\E[XY]$ with $\E[X]\E[XY^2]$ . The ...
-1
votes
1answer
30 views

limit superior and law of large numbers [on hold]

I am wondering whether the following result is true: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ...
0
votes
1answer
27 views

Isometric embedding of $\ell^2$ into $L_1$.

Let $\{Y_n\}_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables on some probability space $(\Omega, \mathcal{F}, P)$ following a standard complex Gaussian distribution (that is, the ...
-1
votes
1answer
18 views

Converse to continuous mapping theorem [on hold]

If $X_n\xrightarrow{d}X$, and $g$ is a.s. continuous, then $g(X_n)\xrightarrow{d} g(X)$. What if we know that $g(X_n)\xrightarrow{d}g(X)$, and $g$ is some continuous function. Can we claim that $X_n\...