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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Expectation and covariance

You have $2$ patches of pieces containing $21$ good and $3$ bad pieces each , you pick a piece randomly from patch $1$ and put it in patch $2$. Let $X$ be the number of bad pieces in the $1$st patch ...
2
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1answer
47 views

Unfair coins connected in a game

I would like to ask the following question. There are 3 coins ($A,B$ and $C$) that are biased with probability of tails equal to $t_a, t_b$ and $t_c$ respectively.   The coins are tossed: ...
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1answer
29 views

Can continuous random variables ever have positive probability on a single point?

From a textbook: Continuous random variables can lead to confusion. First, note that if $X$ is continuous then $\mathbb{P}(X = x) = 0$ for every $x$. But then later: Let $F$ be the CDF for a ...
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12 views

Convergence of a mixed distribution

Let $Y_n=Z+\sum_{i=1}^n \delta_{1/i^2}$, with $\delta$ a point mass and $L(Z)=N(0,\sigma^2)$. Show that $\lim_{n\to\infty} Y_n=Y$, where $L(Y)=N(\frac{\pi^2}{6},\sigma^2)$ The answer file uses ...
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1answer
28 views

Why aren't CDFs left-continuous?

Let $F$ be a cumulative density function on $\mathbb{R}$. From an argument in a textbook, it is shown that $F$ must be right-continuous: Let $x$ be a real number and let $y_1$, $y_2$, $\ldots$ be ...
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44 views

Limit theorem for changed time

This post seems long, but its almost everything proofed except the last step. The unknown part is marked especially. Given a Levy-Process $U_{t}$ with with $E(U_t)=0$ (then $U_t$ is a martingale). ...
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2answers
336 views

Why “One cannot construct more than countably many independent random variables”?

I'm reading the book "Large Networks and Graph Limits" by László Lovász. On the page 18 he said the following: One cannot construct more than countably many independent random variables (in a ...
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25 views

Computing the expected value of $f$ where $\mu$ is a function of $f$

I am curious about the following scenario (This may not be the most sensible set up because I am puzzling this out independent of any homework etc). Suppose $X\sim\mathcal{N}(\mu,\sigma)$ and let ...
0
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0answers
13 views

Kullback-Leibner divergence true distribution

I have an image with an object which I treat as 2-dimensional Gaussian random vector with mean equal to the center of the object surrounded by, roughly, 3-sigma ellipsoid. On the other hand I feed the ...
0
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0answers
19 views

Why does the following equality hold in proving Meyer's inequality?

I have a question in proving Meyer's inequality. The proof I read is taken from the book "Malliavin Calculus and related topics" by Nualart. I just have one equality which I am not sure, I will ...
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0answers
31 views

Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. ...
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1answer
12 views

How the value of denominator calculated here?

I found this example in a book and it has to find probability distribution as stated below: If a car agency sells 50% of its inventory of a certain foreign car equipped with side airbags, find a ...
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22 views

How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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0answers
8 views

Uniform convergence of the action of a Feller semigroup in one variable.

Assume we have two subsets of the some euclidean spaces $X\subset \mathbb{R}^m$ and $Y\subset\mathbb{R}^n$ and a a Feller semigroup $(Q_t)_{t\geq 0}$ on $Y$. Suppose also that we have a continuous ...
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2answers
650 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
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1answer
20 views

Distribution of a transform of bivariate to univariate random variable?

Suppose we have two random variables $$R\sim U[1-\varepsilon,1]\;\;\;\;\; \Theta\sim U[0,2\pi],$$ and a third random variable $$X=g(R,\Theta)=R\cos\Theta.$$ What is the density $p_X(x)$? The ...
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38 views

How can this be a characteristic function if it's not continuous

Consider the following probability density function $f(x)$ \begin{cases} 0 & x<-1 \\ 1+x & z\in[-1,0] \\ 1-x & z\in[0,1] \\ 0 & x>1 \end{cases} Then the ...
2
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1answer
14 views

Using pdf or marginal pdfs to calculate expected value

I have a little doubt concerning the calculation of expected values when dealing with marginal distributions. Consider, for instance, a real bidimensional random variable $(X,Y)$ with pdf $f(x,y)$, ...
2
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0answers
40 views

Is this transformation of a Markov process again Markovian?

Let $(X_t)_{t\in\mathbb{N}_0}$ be a stationary Markov process valued in $\mathbb{R}$ and $c\in\mathbb{R}$. Is the process $(Y_t)_{t\in\mathbb{N}_0}$ defined by $$ Y_t={\bf 1}{(X_t<c)} $$ again a ...
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0answers
15 views

Do characteristic functions characterize the independence of random variables? [Solved] [duplicate]

It is well known that the probability density function characterizes the independence of random variables in the following sense. $$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$ where $f$ is ...
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1answer
21 views

Represent a Uniform[0,1] random variable as a sum of independent Bernoulli(1/2) random variables

With $X \sim U[0,1]$, Lecturer says that $X = \sum_{k\ge 1} B_k(\frac{1}{2}) 2^{-k}$ where the $B_k(\frac{1}{2})$ are independent Bernoulli random variables with parameter $1/2$. I have no idea how ...
2
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0answers
18 views

Esscher Transform extended

The Esscher-transform is a well know tool in the financial section. I posted this in statistics also, since it relates to continuous sampling. Im not sure if my approach is right, so it would be nice, ...
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0answers
15 views

A Stochastic Indicator Function as Wiener Integral

I'm working on a stochastic pde problem and have come across a troublesome source term: Consider $W_t$ a standard Brownian motion. Consider a variable $x \in \mathbb{R}$. Is it possible to rewrite an ...
2
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1answer
40 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 ...
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0answers
16 views

Why is the probability of extinction given by the probability generating function applied to 0?

I am trying to understand branching processes and can't find a good explanation for why solving for the probability of extinction at time $n$ is given by $p^{(n)}(0)$ with the superscript ...
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0answers
25 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 ...
7
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1answer
159 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
2
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0answers
23 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 ...
2
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0answers
43 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 ...
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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13 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 ...
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1answer
36 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, ...
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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 ...
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32 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 ...
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0answers
12 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 ...
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63 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 ...
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0answers
11 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 $(y,\vec{x})$. ...
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1answer
31 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
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0answers
25 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 ...
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1answer
26 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
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1answer
41 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 ...
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1answer
18 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 ...
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1answer
20 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.
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25 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
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1answer
77 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 $h(t)$ ...
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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 ...
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1answer
49 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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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
26 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." ...
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1answer
36 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 ...