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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18 views

Is the density of an absolutely continuous distribution necessarily unique? [on hold]

Is the density of an absolutely continuous distribution necessarily unique?
2
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0answers
19 views

Concentration inequality for sum of squares of independent and identically distributed sub-exponential random variables?

Suppose $X_1, X_2, \ldots, X_n$ are independent and each has the same distribution with a sub-exponential random variable $X$ (for example, $X$ is the square of a standard normal Gaussian variable). ...
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1answer
28 views

probability of not getting same number twice in a row after n die rolls

Having rolled a die $n$ times, I want to determine the probability of not getting any number twice in a row. If I wanted the probability of not getting any number three times in a row, I could use the ...
3
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0answers
34 views

When is the conditional expectation function equal to a continuous function a.e.?

We are given a random variable $Y$ and a $d$-dimensional random vector $X$. Suppose $Y$ is $L_1$ (has first moment). Then $f(x)=\mathbb E[Y\mid X=x]$ is a Borel function. Lusin's theorem says that for ...
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0answers
8 views

Constructing a copula that satisfies the desired condition

Exercise 2.8 in Roger Nelson's An Introduction to Copulas asks the reader to construct a copula $C(u,v)$ not equal to $\max(u + v -1 , 0)$ that satisfies the property $$ C(u,u) = \max(2u - 1,0) $$ ...
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0answers
17 views

Equivalent definition of singular random variable

I'm taking an intermediate course in probability theory (that is without measure theory) and when defining singular random variables (after showing the devil's function), the book defines: $X$ is a ...
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0answers
7 views

Asymptotic distribution for non differentiable functions of estimators

is there kind of a standard tool to derive the distribution of $f(\theta)$ if f is non differentiable (so no Delta Method available) and $\theta$ is asymptotically normal distributed? Thanks a lot!
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8 views

Rigorous Derivation of Metropolis-Hastings Transition Density

The Metropolis-Hastings MCMC algorithm is as follows. Set $X_0$ to some initial value in the support of the target density $f$ and choose a proposal density $q(y \mid x)$; a density in $y$ for each ...
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1answer
23 views

Why is $m(n)\approx\log_2(n)$?

Why is $m(n)\approx\log_2(n)$ ? If $m(n)=\inf\{m:2^{-m}m^{-3/2}\le\frac1n\}$, taking log of $m(n)$ I get $-m(n)-\frac32\log_2(m(n))\le-\log_2(n)$ (This appears in the solution of an exercise in ...
2
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1answer
26 views

$\sigma$-algebra produced by a subclass of a class.

im studying the book 'probability & measure' by Patrick Billingsley. in chapter 2 there's an exercise 2.9 say's: show that: If $B\in\sigma(A)$, then there exists a countable subclass $A_B$ of $A$ ...
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1answer
28 views

Probability of infected but does not show symptoms of disease?

A person moving through a tuberculosis prone zone has a $50\%$ probability of becoming infected. However, only $30\%$ of infected people develop the disease. What percentage of people moving through a ...
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0answers
13 views

What is the rate of convergence in the uniform local limit theorem?

Let $(X_i)_i$ be a sequence of iid random variables, s.t. for some sequences $a_n, b_n$ the normalized sum $$Z_n=\frac{X_1+\dots+X_n}{b_n}-a_n$$ converges weakly to an $\alpha$-stable distributed ...
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0answers
33 views

Is the expected value of a monotone function on a uniformly distributed random variable monotone? [on hold]

Consider the following definition: A sequence of uniformly distributed random variables $(X_n)_{n \in \mathbb n}$ where $X_{n-1} \sim U[a_{n-1},b_{n-1}]$ and $X_n \sim U[a_{n},b_{n}]$ such that $a_n ...
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3answers
15 views

Basic question about the joint probability.

I have a doubt regarding the joint probability. Experiment: I have a deck of cards marked from $1$ to $4$. Two cards are drawn in sequence without replacement. Let $X$ denote the random variable ...
1
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1answer
13 views

Independence of two non-negative integer valued random variables

Let $X,Y$ be two non-negative integer valued random variables defined on a probability space $(\Omega,\cal F, \Bbb P)$. The question is, If $\Bbb P\{X=i,Y=j\}=\Bbb P\{X=i\}P\{Y=j\}$ for every ...
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1answer
22 views

Proving linear operators are Markov Generators

I am trying to do the following question from Liggett. Let $A$ be a linear operator defined on the space of continuous functions $C(E)$ for a compact set $E$. Define $A$ by $A=T-I$ where $T$ is a ...
2
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2answers
28 views

How to prove $P(|X_n-X_m|>\epsilon)\leq P(|X_n-X|>\epsilon/2)+P(|X_m-X|>\epsilon/2)$?

Consider random variables $X, X_1, X_2, ...$ in a probability space $(\Omega, \mathcal F, P)$ such that $X_n\stackrel{p}{\rightarrow} X$. Let $n, m \in \mathbb N$. How can I prove that for some ...
2
votes
1answer
20 views

Lévy Process existence of the expectation of the supremum of the past process.

Given a Lévy Process $X_{t}$ in $\mathbb{R}^{d}$, with $X_{t}^{*}:=\sup_{s\in[0,t]}|X_{s}|$. I want to show, that for $t>0$ with $E[|X_{t}|]<\infty$ for $t>0$, then ...
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0answers
19 views

To estimate the probability that a diffusion reaches a certain value

I have a diffusion process define by the following equation: \begin{equation} dX_t=X_t[\beta(N-X_t)-\alpha]dt+\sqrt{X_t(\beta(N-X_t)+\alpha}) { }dB_t \end{equation} and I proved that the solution ...
0
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1answer
25 views

Variance of Signum Function of Two Random Variables

Let $ X $ and $Y$ be two random variables with means $\mu_X$ and $\mu_Y$ respectively, as well as variances $\sigma_X$ and $\sigma_Y$ (all of which exist). I am interested in computing the following ...
2
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1answer
26 views

Conditional expectation of a random walk given that it is positive

Let $\{\xi_k\}$ is a sequence of iid random variables with $E(\xi_1)=0$ and $E(\xi_1)^2=\sigma^2<\infty$. Define the random walk $Y_n=\sum_{k=1}^n \xi_k$. Is it necessarily true that the ...
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0answers
22 views

How to compute the Lebesgue-Stieltjes measure for given intervals

Let u be a Lebesgue-Stieltjes measure on the Borel σ-algebra. Let Fu be the associated function such that u([a,b)) = Fu(b)−Fu(a). Calculate a) u([a,b]); in terms of the function Fu. b) u((a,b)); in ...
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1answer
35 views

Prove the following integral is asymptotically zero

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show ...
1
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1answer
14 views

Independence: norm v.s. direction of a standard multivariate normal vector

Suppose that $v\sim N(0,\sigma^2 I_n)$ and with $||\cdot||$ denoting the Euclidean norm, define $$ u=v/||v||\quad\text{and}\quad w=||v||. $$ I've been told that $u$ and $w$ are independent and I see ...
1
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1answer
31 views

Showing that $\mu$ is a measure when continuous from above

Statment Let $\mu$ be a set function defined on a $\sigma$ -algebra. Show that $\mu$ is a measure given that $\mu \geq 0$, $\mu(\emptyset)=0$, $\mu$ is continuous from above and countably additive. ...
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1answer
37 views

Poisson probability of an event A before event B

I'm trying to calculate the probability of two poisson processes events happening one before the other, with two different $\lambda$s. The way I see it, I can word it as the probability of event $A$ ...
1
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0answers
39 views
+50

Simple random walk on $\mathbb Z^d$ and its generator

I'm still trying to figure out definitions and properties of random walks on $\mathbb Z^d$. My goal is to work up to understanding some large deviation principles for the local times of such random ...
0
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0answers
10 views

Proof of substitution rule for conditional expectation

Let $v: \mathbb{R}^2 \to \mathbb{R}$ be a function and $X, Y$ random variables. It holds $$ \mathbb{E}[v(X,Y)|Y=y]=\mathbb{E}[v(X,y)|Y=y], \ y\in R(Y). $$ What would be a way to start the proof? I ...
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1answer
14 views

Reformulation of SLLN with continuous nondecreasing process as time

Given a filtered probability space $(\Omega,\mathcal{F}_{t},P)$ and a continuous nondecreasing process $U_{t}$ with $U_{0}=0$ and $U_{t}\rightarrow \infty$ $P-a.s.$ as $t$ goes to $\infty$. Given a ...
0
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1answer
24 views

Finding $P(C)$ with Bayes's Theorem

We have two events $C$ and $D$ such that $0<P(D)<1$ and a $P(C|D)=P(C|D^{c}) = \frac{1}{3}$. I am wondering if it is possible to calculate $P(C)$ from only this information. I've tried using ...
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0answers
12 views

renewal process and jumps(question)

Let $N_t$ be a renewal process and $T_n$ the jumps with $T_n=X_1+...+X_n$.$X_1,..,X_n$ where $X_i$'s are independent random variables identically distributed law $F_X$. Let $A_t:=t-T_{N_t-1}$ and $x ...
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0answers
14 views

Finding the autocorrelation of $X(t)$ and $Y(t)$ from the autocorrelation and pseudocorrelation of $Z(t) = X(t)+i Y(t)$

Consider $Z(t) = X(t) + iY(t)$, $i$ being imaginary. Knowing that $$ r(t_1,t_2) = e^{i(t_1 - t_2) - (t_1+t_2)^{2}} \quad\quad\text{and}\quad\quad \mathrm{pseudo}-r(t_1, t_2) = 0 $$ how can one ...
2
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0answers
29 views
+250

Finding where the tail starts for a probability distribution, from its generating function

Suppose we generate "random strings" over an $m$-letter alphabet, and look for the first occurrence of $k$ consecutive identical digits. I was with some effort able to find that the random variable ...
1
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1answer
43 views

Given three independent events $A,B,C$, is $I_A+2I_B$ independent of $I_C$?

Let $(\Omega,\mathcal{F},P)$ be a probability space and $A,B,C\in \mathcal{F}$ are independent. Is $I_A+2I_B$ independent of $I_C$? $I_A,I_B,I_C$ are indicator random variables. I started by ...
5
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2answers
54 views

Is Probability really consistent with our world?

Say we have 6 unbiased coins, We toss 5 coins and get 5 heads. Then what is the probable outcome of the sixth toss? Mathematically every new and discrete event should be independent of the results of ...
0
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1answer
25 views

How to prove $E\|Y'\|\leq E\|Y'-Y''\|,$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$?

How to prove $$E\|Y'\|\leq E\|Y'-Y''\|,$$ where $Y$ is a random matrix in $\mathbb{R}^{n\times n}$, and $Y'$, $Y''$ are independent copies of $Y$; $\|\cdot\|$ denotes the $l_2$ operator norm;$E$ ...
3
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0answers
56 views

If $X_n\nearrow X$ then $E(X_n)\rightarrow E(X)$?

Let $(X_n)$ be an increasing sequence of real valued integrable rvs on a probability space $(\Omega,\mathcal{F},P)$, such that $(X_n)$ converges ae to some rv $X$. Is it true that $E(X_n)\rightarrow ...
1
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2answers
34 views

If $X_n\geq 0,~X_n\rightarrow X$ ae and $E(X_n)\leq c,$ then $E(X)\leq c.$

Let $(X_n)$ be a sequence of positive valued rvs on a probability space $(\Omega,\mathcal{F},P),$ such that $(X_n)$ converges ae to a rv $X.$ If $E(X_n)\leq c<+\infty$ for all $n$, then $X$ is ...
1
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1answer
25 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
1
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1answer
68 views

If $dX_{t} = X_{t}\,dt + \,dB_{t}$, why does $e^{- t}dX_{t} = e^{-t} X_{t} \,dt + e^{-t} \,dB_{t}$?

I'm taking a course in stochastic differential equations, and in order to solve $dX_{t} = X_{t}\,dt + \,dB_{t}$, the book gives a hint: to multiply both sides of this equation by $e^{-t}$. (But, as ...
3
votes
1answer
44 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
3
votes
0answers
24 views

Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
0
votes
0answers
30 views

Probability Mass Function of a Sentence

We have a sentence: Some dogs are brown. We choose one letter (out of the 16) at random. Let Y be the length of the whole word containing the letter. How can I find the probability mass function of Y? ...
1
vote
1answer
39 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
0
votes
2answers
31 views

Fourier transform problem with symmetric matrix. Related to Gaussian?

Hi everyone I encountered a problem that looks simple enough but I have no idea where to start. Find Fourier transform of $e^{-\langle Ax,x\rangle}$ when $A$ is a positive definite symmetric $n ...
1
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0answers
27 views

The “how many pieces do you have buy on average” problem, a markov problem?

I recently discovered a problem similar to this one in a book about Markov chains: Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the ...
0
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0answers
34 views

Are there different definitions of a continuous time Markov chain, condition on a finite or infinite number of earlier values?

My book defines a continuous time Markov chain like this. Let $\{X_t\}, t \in T$ be a stochastic process on $(\Omega, \mathcal{A}, P)$, with a countable state space $S$. The process is a Markov ...
0
votes
1answer
43 views

Conditional entropy and independent conditioning variables

Let $X,Y,Z,Y',Z'$ be random variables where $Y\sim Y', Z\sim Z'$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are, in the sense that we have $p(X,Y,Z)=p(X|Y,Z)p(Y)p(Z)$ ...
1
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0answers
34 views

If $P(X_1 < X_2)$, what is $P(X_1 < X_2 \cap X_1 < X_3)$?

Say $X_i$ can have a real value in the range [1,100]. All $X_i$ are independent of each other and all values are equally likely. So then $\mathbb{P}(X_1 < X_2) = \frac{1}{2}$, right? But then, ...
0
votes
1answer
23 views

uniform Distribution on uncountable Lebesgue $0$-Sets

I know that for every measurable Set A it is possible to create a uniform Distribution on A if - A is finite - A is not a lebesgue 0-set and its not possible for infinite countable sets so I ...