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 ...

learn more… | top users | synonyms (1)

0
votes
0answers
11 views

Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
0
votes
2answers
26 views

Correct notation for Continuous random variables

Assume two random variables $X \in \mathcal{X}$ and $Y \in \mathcal{Y}$. Then, if $X$ and $Y$ are discrete random variables $P(X \in E) = \sum_{y \in \mathcal{Y}} P(X \in E, Y=y)$. I want to know ...
0
votes
0answers
10 views

Does Strong Markov property need time homogenous property?

If $X$ is a Markov chain, we know that for any bounded measurable function $f$, $E[f(X_{n+1}) \vert \mathcal{F}_n] =E[f(X_{n+1}) \vert \sigma(X_n)] = g(X_n)$ where $\sigma(X)$ is the sigma-algebra ...
0
votes
0answers
12 views

How can I calculate definite integral of chi-squared pdf with one degree of freedom

enter image description here I need a calculating process of the above definite integral please help me.. (sorry for my poor English)
2
votes
2answers
16 views

$X_n\to 0$ in probability implies $E[f(X_n)]\to f(0)$ for $f$ uniformly cts and bounded

Let $f$ be a uniformly continuous and bounded function. I've shown that if $X_n\to 0$ in probability, then $f(X_n)\to f(0)$ in probability as well. Now I want to say that $$\lim_{n\to\infty} ...
1
vote
0answers
21 views

Specific Radon-Nikodym Derivative Interpretation

Suppose $(\Omega, \mathcal{F}, P)$ and $(\Omega, \mathcal{F}, Q)$ are two probability spaces. The Radon-Nikodym theory says there exists a function $f$ on $\Omega$ that satisfies $$ P(A) = \int_A f ...
0
votes
0answers
8 views

Total variation relates to a norm: how about some metric inducing the weak topology?

Let $\mu,\mu'$ be Borel probability measures on a Polish space $S$. There's a straightforward definition of the signed measure $\mu-\mu'$ and on the space of signed measures, we have the total ...
2
votes
1answer
25 views

Pdf of $Z=X/Y$ given a joint pdf

Find the p.d.f. of $z=x/y$ $f(x,y)= 2(x+y)$ for $0\le x\le y\le1$ I first did the simple way, transformation, then derivative, and multiply joint p.d.f by absolute value of the derivative. Then ...
-2
votes
0answers
25 views

How is Mathematics and Space related? [on hold]

I've seen scientists come up with equations and proofs in mathematics of some incident happened or happening in space. How do they relate mathematics with that? For Example How do you start ...
1
vote
0answers
16 views

Uniqueness of Predictable Quadratic Covariation

In order to prove that $\langle M,N \rangle$ is the only process which is continuous and has bounded variation such that \begin{align} M_tN_t - \langle M,N \rangle_t \end{align} is a continuous ...
0
votes
1answer
14 views

Sufficient conditions for a probability measure to be characterized by finite number of moments

Suppose $Y(X)$ is a random variable defined on the probability measure $X$ with density, $f$, with respected to the Lebesgue measure and distribution function $F_y$. Consider the following ...
2
votes
2answers
32 views

Is covariance preserved under transformation?

Let $X_1,X_2$ be normally distributed random variables with $\rho = 0.5$, mean equal to $0$ and variance equal to $1$. Let $U_i = \Phi(X_i)$ where $\Phi$ is the marginal distribution of $X_1,X_2$. We ...
0
votes
1answer
6 views

Sufficient Estimator How to proceed?

Let $X$ be a random variable with exponential density function. Show that the mean of $X$, denoted $\overline{X}$, is the sufficient estimator of $\lambda$ but not an unbiased estimator of $\lambda$
0
votes
1answer
29 views

Prove $X_{\infty} < \infty$

From Williams' Probability with Martingales: How exactly do we prove $X_{\infty} < \infty$ a.s.? $$E[|X_{\infty}|] = E[|\lim X_n|] = E[|\liminf X_n|] = E[\liminf |X_n|]$$ $$ \le \liminf ...
2
votes
0answers
15 views

Characteristic functions proof… $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$

I am trying to prove Lemma 4.1 (1) from Olav Kallenberg's Foundations of Modern Probability: $\mu\{x:|x|>r\}\le\tfrac r2\int_{-2/r}^{2/r}(1-\varphi(t))\,dt$ From context, I believe that ...
1
vote
1answer
25 views

Sequences of random variables converging in probability to the same limit a.s

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...
2
votes
0answers
21 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
1
vote
1answer
23 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
0
votes
1answer
28 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
0
votes
1answer
17 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
0
votes
0answers
14 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
0
votes
0answers
14 views

Clarifications on proof of Doob's Forward Convergence Theorem, warning related to it and proof of a corollary

From Williams' Probability with Martingales: $X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$ --> Is this supposed to be stronger than $\lim X_n$ does not exist because it's ...
1
vote
0answers
32 views

Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
1
vote
0answers
31 views

Integrals with erf^N

Can anyone help with integral of type. In general, what to do if erf is in power higher than 1? $$g(S|S<L)=\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{+\infty} \left [ \frac{1}{\sqrt{2 \pi ...
0
votes
1answer
18 views

Given Conditional Expectation check if it satisfies the density function. [on hold]

I have this math problem that I can't seem to solve. Let $X$ be a continuous random variable that only takes non-negative value that satisfies $\mathbb{E}(X\mid X \ge t) = t + ...
0
votes
1answer
31 views

Does finite expectation imply finite essential supremum?

I have a real valued function $f$ with the property that $$\mathbb{E}\big[f(X)\big] = \int f(x)\ d\mathbb{P}(x) \leq c$$ for some $c > 0$. Does this imply $$ \operatorname{ess sup } \|{f(X)}\|^2 ...
0
votes
1answer
21 views

Is sharing the same support a necessary condition for exchangeability?

I am confused on the meaning of exchangeable random variables. The question is: consider the random variables $X_1,X_2,X_3$ defined one the same probability space $(\omega, \mathcal{F}, P)$; is ...
1
vote
0answers
28 views

Expectation formula

Let $F(z)=P\{ Z \leq z\}$. Assume $F(c)=0$. It is well known that: $$E(Z)=\int_c^{\infty}(1-F(z))dz$$ and more generally: $$E(g(Z))=\int_c^{\infty}g(z)dF(z)$$ Is it also true for: $\tilde{F}(z)=P\{ Z ...
0
votes
1answer
19 views

Density of a distribution given by a Gaussian copula and a set of marginals

Suppose the distribution of an $n$-dimensional random vector $X$ is characterized by a Gaussian copula $C_R$ with correlation matrix $R$ and a set of marginal $\{(F_{X_i}, f_{X_i})\}_{i=1}^n$ (pairs ...
1
vote
0answers
32 views

Riemman-Stieltjes Integral Exercise

The truth is that I have no experience with the integral of Riemann-Stieltjes and developing a Bayesian inference problem in the book "Mathematical Statistics" by Shao, appears one of these steps, I ...
1
vote
0answers
21 views

Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
1
vote
1answer
22 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
0
votes
2answers
33 views

Probability of a number in the real line

I have read that the probability to pick a rational number in the real line is null. My problem is: If $S$ is a dense set in the real line, what is the probability to pick an element of $S$? There ...
0
votes
1answer
24 views

$0 \leq Y \leq M$ random variable, $p > 1$. Calculate $\mathbb{E}(Y^p)$

$0 \leq Y \leq M$ random variable, $p > 1$. Show that $\mathbb{E}[Y^p] = \int_0^M py^{p-1}\mathbb{P}[Y \geq y] dy$ My attempt: $\mathbb{E}[Y^p] = \int_0^{\infty} Y d\mathbb{P} = \int_0^{M} Y ...
0
votes
1answer
23 views

Is this martingale constant 0?

I have a martingale X where $X_0 = 0$ a.s. And for each $\omega$, the path $f(t)=X_t(\omega)$ is of bounded variation in the classical sense. That ...
1
vote
0answers
49 views

On a Probability notation - $\mathbb{E}[X(.)|\mathcal{F}]_G$

What could mean this notation : $\mathbb{E}[X(.)|\mathcal{F}]_G$ ? where G : $\Omega \rightarrow \mathbb{R}$ is a random variable on a probability space $(\Omega,P, \mathcal{F})$. X could be a ...
-1
votes
0answers
14 views

Rejection method in R

I need to generate pseudo random numbers using the rejection method in R X is a random variable normally distributed N(6,1). I did as follows, but I get a error message. Can someone give me a hint? ...
0
votes
0answers
20 views

Expected Value of $P(|Y_n^{(K)}| > \epsilon)$ where $Y_n^{(K)}$ is the random sum of a sequence of RV converging to 0 in Probability

I have been struggling with this for countless hours, I would appreciate a hint to get me going in the right direction (no complete answer please) Problem: Assume that for all $k \in \mathbb{N}$ ...
2
votes
1answer
41 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
0
votes
0answers
21 views

Distribution of convex combination of Bernoulli random variables

Suppose $Y_1,Y_2, \ldots $ are i.i.d Bernoulli$(p)$. What is the distribution of $$\sum_{i=1}^{\infty} \frac{Y_i}{2^i}$$ I could deal case for $p=\frac{1}{2}$ using characteristic functions but for ...
0
votes
0answers
19 views

A problem on random q-colourings of a graph for randomly chosen vertex

Here is an exercise from Olle Haggstrom's "Finite Markov Chains and Algorithmic Applications" from the chapter "Fast Convergence of MCMC Algorithms". The exercise is based on random $q$-colorings of ...
3
votes
1answer
32 views

$xP(|X|>x) \rightarrow 0 $ implies $E|X|^{1-\epsilon} < \infty$

Given $X$ is a random variable, and $\epsilon \in (0,1)$, prove that: $$xP\left(|X|>x\right) \to 0 \quad \text{implies} \quad \Bbb E|X|^{1-\epsilon} < \infty$$ I got a hint to use the ...
1
vote
1answer
20 views

Distribution of $aXa^T$ for normal distributed vector $a$

Let $a$ be $1\times n$ random vector with entries chosen independently from normal distribution with zero mean and unit variance. What is the distribution of $aXa^T$ for a given $n\times n$ matrix ...
0
votes
2answers
30 views

Probability that digits 1,2 and 3 will appear in a decimal m digits, how do I tweak my thinking to be correct?

So I first thought to approach this as the complement of an inclusion exclusion problem, $P(A_{1}\cap A_{2}\cap A_{3})=1-P(A_{1}\cup A_{2}\cup A_{3})$ Where $A_{i}$ is the event that digit i appears ...
1
vote
1answer
21 views

Fact check probability De Morgan's Law

Is the statement about the De Morgan's Law application correct? Isn't there a ''c'' (compliment) too many?
2
votes
0answers
44 views

Distribution of $1/U^2$ where $U$ is uniformly distributed on $(-1, 1)$

Suppose $U\sim \mathrm{Uniform}(-1, 1)$. Let $Y =1/{U^2}$. What is the distribution of Y? Here is what I have: $$ \begin{aligned} Y \in [1,\infty)\\ P(Y <y) = P\Big(\dfrac{1}{U^2} < y ...
-2
votes
0answers
23 views

Length of shorter arc of a circle cut at two random points [on hold]

two points are chosen uniformly and independently on the perimeter of a circle of radius 1. This divides the perimeter in two pieces. Determine the expected value of the length of the shorter piece
0
votes
0answers
17 views

Pure Birth Question. Find the probability that the population at time $t$ is an odd # given it starts at $0$.

Here is the question. Consider a pure birth process $\{X(t) : t ≥ 0\}$ with birth parameters $\lambda_{2n} = α>0$ and $\lambda_{2n+1} =β>0$ for $n∈N$. Compute $Pr\{X(t) \text{ is odd } \mid ...
1
vote
0answers
42 views

Friedman's urn is a supermartingale or a submartingale?

Here is the urn model: At time zero there are $r$ red and $g$ green balls in an urn. At each time-step, we draw out a ball at random and replace it along with $c$ of the same color and $d$ of the ...
0
votes
1answer
31 views

Shape of distribution between arrivals in a poisson process

Note that both diagrams are refferring to the same problem. The difference is that I'm not sure if graph I'm supposed to visualize is the PDF or the CDF, so I drew them both and hope someone will ...