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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Proof of linearity for expectation given random variables are dependent

The proof of linearity for expectation given random variables are independent is intuitive. What is the proof given there they are dependent? Formally, $$ E(X+Y)=E(X)+E(Y)$$ where $X$ and $Y$ are ...
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39 views

Strong existence of solutions to SDE and continuity in the time variable

I recently come across some literature in stochastic analysis that uses the following result: Consider the one-dimensional SDE $$dX_t= a(t, X_t) \, dt + b(t, X_t) \, dW_t, $$ where $a, ...
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2answers
43 views

Theorem on intersection of events using conditional probability

I understand that $$P(A,B)=P(B)P(A\mid B)=P(A)P(B\mid A)$$ but the generalization of it is a bit confusing to me. $$P(A_1,A_2,\ldots,A_n)=P(A_1)P(A_2\mid A_1)P(A_3\mid A_2,A_1)\cdots P(A_n\mid A_1,\...
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26 views

“Uniformly choosing” from $\mathbb Q\cap(a,b)$

Take the set of rational numbers $Q=\mathbb Q\cap(a,b)$. Step 1: Start with an arbitrary sequence $Q_0=\left\{q_1, q_2,\ldots\right\}=Q$ where each rational number appears exactly once. Step 2: ...
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27 views

Poisson Process Exercise

A flashlight needs two batteries to be operational. Consider such a flashlight along with a set of $n$ functional batteries — battery $1$, battery $2$, $\ldots$ , battery $n$. Initially, battery $1$ ...
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95 views

When is the compensated Poisson random measure a martingale ? (extensions to sets not bounded from 0)

Assume you have a Lévy process X. Let $N(t,A)$ be defined as the number of jumps in the interval $(0,t]$, such that the jumps size $\Delta X_s \in A$. It can be shown that if $0 \ne \bar{A}$, then $...
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12 views

Econometrics, forecast, in sample and out of sample regression

Is there anyone who can explain me the difference between an in sample regresion, out of sample expansion regression and an out of sample rolling regressions? I have time series data from 1990 to 2016/...
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55 views

Convergence of a process

this may be viewed as a duplicate of this post. However i have put in much effort in the shared link and donated it with reputation, to check the proof considered there. Here however i want to argue ...
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1answer
89 views

Average number of terms required in a sum of exponential variables to reach a specific limit

I have a sum $Y=\sum_{i=1}^{\infty}(X_i-t)u(X_i-t)$ where all $X_i's$ are i.i.d exponentially distributed random variables with parameter $\lambda$ and $t$ is a constant. I want to know how many term ...
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9 views

Infinite differentiability of a (cumulant generating) function

Suppose that I have a moment generation function $M(t)$ that is positive and finite for all $t\in\mathbb{R}$. I know that this implies that $M(\cdot)$ is infinitely differentiable on $\mathbb{R}$. ...
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58 views

Integral of conditional probability density function

As far as I understand, when we fix the condition for the conditional density, we get probability distribution and the integral over all the space is $1$ $P(X|Y=y_0)$: $$\int_{\mathbb{R}}f_{X \mid Y}(...
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1answer
48 views

Why does the pmf for a Poisson Distribution Maximize at $x = \lambda$?

For a random variable $X$ s.t. $X$ has a Poisson distribution: $$ P(k \text{ events in interval}) = \frac{\lambda^k e^{-\lambda}}{k!} $$ The following graph seems to indicate that the maximum ...
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14 views

Bayesian equation: need for priors

As far as I understand, in the problem of Bayesian inference we have a random variable $y$ describing data, which is distributed according to some parameter $x$ via the known conditional distribution $...
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23 views

Probability reference after Billingsley's Convergence of Probability Measures

I've read through Durrett's Probability: Theory and Examples and Billingsley's Convergence of Probability Measures and was wondering what a good next step would be. I've taken graduate courses in ...
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34 views

Is Bayesian Association mathematically rigorous?

Introduction. This question is based on the Ph.D. thesis of B.T. Vo, which can be found in this website ("Papers" section). More specifically, in the introduction of the Ph.D. thesis, at page 8, there ...
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1answer
34 views

How does a Markov process inherit its homogeneity to the embedded Markov chain?

A homogenous Markov process $\lbrace X(t),t\geq 0\rbrace $ is given and the embedded Markov chain $Y_0,Y_1,\ldots$ is defined as $Y_n:=X(T_n)$, where the $0=T_0<T_1<\ldots$ are the moments where ...
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45 views

Time scaling birth process in Poisson process

Given a birth process $\{B_t:t\geqslant0\}$ with $\lambda >0$, define $$K_t=\int_{0}^{t}B_s ds=\sum_{i=1}^{n}B_{t_{i}}(t_{i+1}-t_i)$$ if there were $n$ births in $[0,t]$ and let $t_{i}$ be the ...
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15 views

Started Counts method for estimating transition probabilities of a discrete time markov chain

I would be very pleased if you could help me with a problem I'm having for my Bachelor's thesis. I'm working on some inventory forecasting methods and one of the method's I'd like to apply is a method ...
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0answers
16 views

Equation with the expectation of a assessed Markov process

In my book about Markov processes there is following equation in a proof and I don't see why it's right, I already ask some people in the university, but I had no success, can somebody help me? $$E(\...
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1answer
35 views

If $Var(X)\leq 1/2$, prove that $P(E(X)-1\leq X\leq 2E(X))\geq 1/2$.

Let $X$ be a positive valued random variable, such that $Var(X)\leq 1/2$ and $E(X)\in \mathbb{R}$. Prove that $$P\big(E(X)-1\leq X\leq 2E(X)\big)\geq 1/2.$$ I was thinking of the ...
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18 views

Contradiction in the CDF derivation from two different strategies

I have a sum $Y=X_1u(X_1-t)+\cdots X_Nu(X_N-t)$ where all $X_i's$ are i.i.d with exponential distribution with parameter $1$ and $u(x)$ is the unit step function. As can be seen from the expression of ...
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2answers
28 views

Right limit of integral

Following is the joint PDF of RV $X,Y$ and $Z$ $$f(x, y, z) =\begin{cases} kxy^2z;& 0 < x,y < 1, 0 < z < 2,\\ 0,& \text{elsewhere}. \end{cases}.$$ To find value of $k$ I tried ...
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2answers
50 views

CDF of sum of N exponentially distributed random variables with condition

I have $Y=X_1u(X_1-x_{th})+X_2u(X_2-x_{th})+\cdots+X_Nu(X_N-x_{th})$, with all the $X_i\sim\lambda e^{-\lambda}$, $u(t)$ is the unit step function and $x_{th}$ being the threshold which means that any ...
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1answer
32 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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19 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
34 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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30 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 $f:...
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18 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 ...
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27 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 ...
4
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1answer
148 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
3
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1answer
86 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
14 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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50 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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13 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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41 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
15 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 $f(x,y)$, ...
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0answers
55 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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47 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
28 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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71 views

Esscher Transform extended

This problem is almost solved, dont get scared by the massive text The Esscher-transform is a well know tool in the financial section. I posted this in statistics also, since it relates to continuous ...
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16 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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21 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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1answer
32 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 ...
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19 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 ...
2
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1answer
44 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
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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1answer
19 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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1answer
42 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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13 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 $(y,\vec{x})$. ...
2
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0answers
27 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 ...