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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2
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1answer
536 views

Superharmonic function and super martingale.

The definition (from Durrett's "Probability: Theory and Examples"): Superharmonic functions. The name (super martingale) comes from the fact that if $f$ is superharmonic (i.e., f has continuous ...
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0answers
8 views

Random Variable with characteristic function cosine

So I am searching for a Random Variable $X$, such that $\varphi_X(t)=cos(t)$. I know how to choose $X$ such that $\varphi_X(t)=e^{it}$ and $\varphi_X(t)=e^{-it}$. Does this help me? How can I put ...
-1
votes
1answer
29 views

Proof of the existence of $E(X|\mathcal{G})$

I am looking through my lecture notes, which follows Billingsley, regarding the proof of the existence of $E(X|\mathcal{G})$. The theorem is: Let $(\Omega, \mathcal{F}, P)$ be a probability space, ...
1
vote
1answer
18 views

Calculate minimal Variance

My task is to calculate the minimal variance. I got a result, but don't know for sure if it's correct. Maybe some of you could help me out here. Let $X$ be some real-valued random variable. We know ...
1
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0answers
12 views

Conditional Radon-Nikodym and disintegration

Here (p. 15) the author defines conditional divergence as $$D(P_{Y\mid X}\mid\mid Q_{Y\mid X}\mid P_X):=\mathbb{E}_{x\sim P_X}\left[D(P_{Y\mid X=x}\mid\mid Q_{Y\mid X=x})\right]$$ for two ...
-1
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0answers
20 views

Why is $P(A|\mathcal{G})$ $\mathcal{G}$-measurable? [on hold]

I want to show that $E(1_{A}|\mathcal{G})=P(A|\mathcal{G})$, for every $A \in \mathcal{F}$, where $1_{A}$ is the indicator function and $\mathcal{G}$ is a $\sigma$-subfield of the $\sigma$-field ...
0
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0answers
9 views

condition for recurrence of semi-stable process

Let $(X_t)$ be a nontrivial $\alpha$-semi-stable process on $\mathbb R$. I want to prove that if $1\le\alpha\le2$, $(X_t)$ is recurrent if and only if it is strictly $\alpha$-semi-stable. I want to ...
8
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0answers
50 views
+100

Exploiting the Markov property

I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs. Let ...
0
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0answers
9 views

What are the differences between stochastic v.s. fixed regressors in linear regression model?

If we have stochastic regressors, we are drawing random pairs $(y_i,\vec{x}_i)$ for a bunch of $i$, the so-called random sample, from a fixed but unknown probabilistic distribution $(y,\vec{x})$. ...
1
vote
1answer
30 views

Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1/2.$$ ...
2
votes
0answers
12 views

Levy process measurable past

For a Levy-process $(X_t)_{t\geq 0}$ with stationary indepedent increments which is a markov process, we know that its law is defined by its one dimensional distribution. This is so because for its ...
1
vote
1answer
19 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) ...
2
votes
1answer
40 views

Probability of picking marbles from a bag with only the ratio of marbles given

Here is a question that is puzzling me: A bag contains a large number of marbles; the numbers of the red, blue and yellow marbles are in the ratio $3:4:5$. Four marbles are randomly drawn ...
1
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1answer
17 views

laplace transform probability distribution not concentrated on 0

This seems intuitively obvious but how to prove that $\hat{\mu} < 1,$ when $\theta >0$ and $\mu$ is a probability measure not concentrated at $0,$ where $\hat{\mu}$ is defined as below ...
0
votes
1answer
18 views

For convex $f$, why is $(p,q) \mapsto q \, f(p/q)$ convex on $\mathbb{R}_+^2$?

This fact was stated in the Wikipedia article on $f$-divergences to explain why they are jointly convex.
2
votes
0answers
20 views
+50

How to make a probabilistic sense of the semigroup of a positive operator

Consider the operator $\mathcal{L}$ acting on the function $f:\{0,1\}\mapsto \mathbb{R}$ defined as following: $$\mathcal{L}f(x)=f(1-x)-f(x)$$ This is the infinitesimal generator of a continuous time ...
2
votes
1answer
74 views

Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
1
vote
1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
3
votes
1answer
38 views

If $\lim\limits_{A \rightarrow \infty} \sup\limits_{n} \frac{\int_{|x|>A}x^2 dF_n(x)}{\int_\mathbb Rx^2 dF_n(x)}=0$ then $\{F_n\}$ is tight

Suppose $X_n$, $n \geq 1$, are random variables with distribution functions $F_n$ satisfying $EX_n^2 < \infty$ for all $n$ and $$\lim_{A \rightarrow \infty} \sup_{n} \frac{\int_{\{x: ...
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votes
0answers
14 views

Tail bounds for functions of a Poisson point process

A Poisson point process consists of a sequence of points $0\leq t_1\leq t_2<\cdots$ where $t_i = t_{i-1} + X_i$ where $X_i$ is an exponentially distributed random variable with some rate parameter ...
0
votes
0answers
18 views

$e^{\varphi -1}$ characteristic function

So I am trying to figure out whether $e^{\varphi-1}$ is a characteristic function given that $\varphi$ is. I know that linear combinations of characteristic functions and the real part of a ...
0
votes
1answer
29 views

Compute E(X|Y) when P(Y=2)=1

I have a problem and don't understand a lot of stuff in there. First, it is given that if there are random variable X and Y where X has a well defined finite expectation, the, $\mathbb{E}(X|Y)$ ...
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0answers
25 views

Given X and Y ind. rv's, when is f(X,Y), g(X,Y) ind.?

I have to parallel questions. I was trying to solve this one: "Given two independent real-valued randomvariables X and Y defined on the same sample space, is it true that X and X+Y are independent." ...
1
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1answer
35 views

The Second Hearts Problem

According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is ...
4
votes
1answer
29 views

Questions about the distribution of $Y$ given the distributions of $X$ and of $Y$ conditionally on $X$

$\newcommand{\Var}{\operatorname{Var}}\newcommand{\E}{\operatorname{E}}$Given: $X$ uniform on $(0,1)$ and $Y\mid X=x$ with distribution $N(x,1)$. Question 1: Determine $\E(Y^2)$ and $\Var ...
1
vote
1answer
23 views

On characteristic function of multivariate normal distribution

I cant make sense of (4.2), putting $Z=1$ should only establish one point not the whole function right, hence I dont see how Thm 4.1 is "proved"? Second RHS is not even a funtion of $t$. Anyone have ...
5
votes
1answer
38 views

Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk ...
1
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0answers
25 views

Statement of the strong Markov property in Norris' book

In J.R.Norris' Markov chains book, the strong Markov property for discrete-time, Markov chains is stated and proved as follows: Let $(X_n)_{n \geqslant 0}$ be a Markov chain with transition ...
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0answers
25 views

Nonanticipativity constraint (filtration/measure theory)

I am trying to show that stochastic process must attend the nonanticipativity constraint using filtration in measure theory. Adaptability of a stochastic process tell us that: $$\sigma(X_t)\subset ...
0
votes
1answer
36 views

Is Littlewood's law true?

Littlewood's law states that a person can expect to experience an event with odds of one in a million (defined by the law as a "miracle") at the rate of about one per month. This is attributed to the ...
2
votes
1answer
67 views

Does $E[e^{it(aX + bY)}]=E[e^{itaX}]E[e^{itbY}]$ for every $a,b\in\mathbb{R}$ imply that $X$ and $Y$ are independent?

Let $X, Y$ be two random variables such that for every $\alpha, \beta \in \mathbb{R}$, $$E[e^{it(\alpha X + \beta Y)}]=E[e^{it\alpha X}]E[e^{it\beta Y}]$$ for all $t\in\mathbb{R}$. Does it follow ...
4
votes
1answer
169 views

Can two random variables $X,Y$ be dependent and such that $E(XY)=E(X)E(Y)$?

Can someone define independence of two random variables with this "product rule", or are there any counterexamples?
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0answers
12 views

Showing that the probability measure is a continuous function for a series of decreasing events [on hold]

) I want to show that given the events $A_1, A_2, \ldots$ are decreasing, that $P(lim_{n \to \infty} A_n) = lim_{n \to \infty}P(A_n)$. Any ideas will be greatly appreciated!
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0answers
19 views

Find probability dense function of multiple random variable

Suppose we have random variable such that $S=g(X_{1},X_{2},...,X_{n})$ and $X_{i}, 0\leq i\leq n$ are all independent and uniformly distributed. I have done my best to find the cumulative distribution ...
2
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0answers
35 views

How to generate correlated random numbers with specific distributions?

After read the answers of some similar questions on this site, e.g., Generate Correlated Normal Random Variables Generate correlated random numbers precisely I wonder whether such approaches can ...
0
votes
2answers
32 views

On the proof that every positive continuous random variable with the memoryless property is exponentially distributed

The theorem to prove is: $X$ is a positive continuous random variable with the memoryless property, then $X \sim Expo(\lambda)$ for some $\lambda$. The proof is explained in this video, but I will ...
1
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0answers
38 views

Bayes' Theorem and Law of total propability for CDF

The calculation of conditional probability is the same for conditional PDF and CDF(according to a number of questionable sources: first, second) (I will use rough notation, with just $x$ and $y$): ...
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0answers
16 views

Difference equations and the characteristic polynomial

The context for this is solving the gambler's ruin problem using linear algebra. I haven't found a good explanation for why the linear combination of the eigenvalues for the matrix representing a ...
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votes
3answers
69 views

Is probability function for mutually exclusive events a linear operator?

If the definitive classification criteria for a linear operator are given by: $L(f+g) = L(f) + L(g)$ [for any/every pair of functions, $f$ and $g$] $L(tf) = tL(f)$ [for ...
1
vote
2answers
335 views

Meaning of a probability distribution being dominated by a measure

The following comes from Ghosh & Ramamoorthi (2003) Bayesian Nonparametrics. In terms of notations, $\Theta$ is a parameter space with Borel $\sigma$-algebra $\mathcal B(\Theta)$. For ...
2
votes
2answers
45 views

An elevator containing five people can stop at any of seven floors.

An elevator containing five people can stop at any of seven floors. What is the probability that no two people get off at the same floor? Assume that the occupants act independently and that all ...
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0answers
23 views

Generalized inverse of a function

It is well-known that if a function is strictly increasing, then it has an inverse function. I also see the concept of "generalized inverse" in the litarature, which has the definition ...
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2answers
28 views

Finding the distribution of a n tossed fair coin

I am trying to solve the problem: Consider a sequence of n tosses of a fair coin. Let X denote the number of heads, and Y denote the number of isolated heads, that come up. (A head is an ...
1
vote
2answers
33 views

A group of 60 second graders is to be randomly assigned to two classes of 30 each…

A group of 60 second graders is to be randomly assigned to two classes of 30 each. Five of the second graders, Marcelle, Sarah, Michelle, Katy, and Camerin, are close friends. (a) What is the ...
2
votes
0answers
29 views

Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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2answers
52 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
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vote
1answer
27 views

A version of one-sided Chebyshev's inequality

Let $X$ be a real random variable with mean $\mu > 0$ and variance $\mu^2$. Does there exist a non-trivial upper bound on the probability $\Bbb P(X < 0)$ or is there a counterexample that shows ...
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0answers
10 views

Distributions of components to distribution of vector

Suppose that I have independent variables $x_1,\ldots,x_n$ with tractable (not necessarily identical) distributions. I'm interested in the distribution of $\boldsymbol{x}=(x_1,\ldots,x_n)'$ and, if ...
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0answers
26 views

Reference for function being a distribution function

Let $\mathcal B(\mathbb R)$ denote the Borel $\sigma$-algebra on $\mathbb R$. Let $\nu$ be a probability measure on $(\mathbb R, \mathcal B(\mathbb R))$. Lastly let $F:\mathbb R \rightarrow [0,1]$ ...
2
votes
2answers
118 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. ...