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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6 views

What is the intuitive difference between almost sure convergence and convergence in probability?

It is a standard fact in probability that almost sure convergence is weaker than convergence in probability. I can only see the differences in the proof. However, is there a way to view it ...
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1answer
9 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
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10 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
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16 views

Probability to get from A to C.

There has been a snowstorm and Bob is trying to drive from A to C. p and q are the probabilities that the two roads are passable. What is the probability that Bob can get from A to C? Note that ...
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1answer
10 views

Stopping time and the Martingale stopping theorem.

According to the book that I am reading, A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of ...
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24 views

Learning the geometry of spaces of probability distribution as fast as possible

Strange statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence or max/minimization with respect to these quantities are ...
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0answers
12 views

Mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$?

Is the mutual information $I((X,Y,Z);A)$ larger for small pairwise mutual informations $I(X;Y), I(X;Z), I(Y;Z)$? In particular, in the extreme case that the pairwise mutual informations are ...
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14 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
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0answers
15 views

Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
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1answer
22 views

Weak convergence of probability measures and uniform convergence of functions

I am stuck on Problem 4.12 of Karatzas and Shreve's book Stochastic Calculus and Brownian Motion: Suppose that $\{ \mathbb{P}_n \}$ is a sequence of probability measures on $(C[0, \infty), ...
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1answer
19 views

Find the joint probability density function of Max and Min

This is the problem 1.2.13 of Karlin's book An introduction to stochastic modeling: Let X and Y be independent random variables each with the uniform probability density function ...
2
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0answers
6 views

(joint) Functional CLT for partial sums and counting process

Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$. Define the diffusion-scaled partial sum as ...
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0answers
17 views

Question on a proof involving tightness and almost sure convergence of a sequence

I'm having a hard time understanding the proof of Lemma 17 in this article. Essentially, the assertion of the lemma boils down to replacing a constant in a sequence of random variables that satisfies ...
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0answers
9 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
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0answers
21 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
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0answers
16 views

What is the probability of unions of intersections?

Suppose we have two unions of (possibly overlapping) events. Let me denote the unions as: $$A = IE_A^1 \cup \dots \cup IE_A^{k_A}$$ $$B = IE_B^1 \cup \dots \cup IE_B^{k_B}$$ Each $IE_X^y$ is a ...
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0answers
24 views

Reflection principle for simple random walk

Let $(X_n)$ be a sequence of independent random variables, such that $P(X_i=1) = P(X_i=-1) = 1/2$. Then, the reflection principle states that for all $a > 0$, $$P(\max_{1\leq k\leq n} S_k \geq a) ...
1
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1answer
13 views

What is the distribution of $\min\limits_{1\le i\le N}\frac{X_i}{Y_i + C}$ as $N \rightarrow \infty$?

Let $Z_i = \frac{X_i}{Y_i + C}$ with $i = 1, 2, \dotsc, N$ denote the sequence of the random variables, where $X_i$ and $Y_i$ are exponentially distributed independent random variables with different ...
2
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0answers
19 views

Conditional expectation of another expectation expression.

What is the intuition and the proof behind the given below expression where $U,V,W$ are random variables: $E[V | W]$ = $E[E[V | U,W] | W]$ I know that $E[V | W]$ can be treated as a random variable ...
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1answer
20 views

proof for the probability$P(A^{c}\cup C)$.

Let $A,B,C$ be pair-wise independent (like A and B are independent) such that$P(A^{c}\cap B)=0.1$ and $P(B\cap C)=0.2$ Show that $P(A^{c}\cup C)\ge \frac{7}{8}$. $P(A^{c}\cup C)=1-P(C^{c}\cap ...
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0answers
9 views

independence of chi square distributions

We already knew that if two independent chi-squared random variables, then their sum is also chi-squared with the degree of freedoms is the sum of theirs. How about the converse? If $X\sim\chi^2(n)$ ...
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1answer
43 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
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1answer
19 views

Finding out the constant term in cumulative distribution function.

Let$$F(x)=\begin{cases} 0,\ if\ x<0\\ \frac{x^2}{10},\ if\ 0\le x<1\\\frac{x+2}{8},\ if\ 1\le x<2\\ \frac{c(6x-x^{2}-1)}{2},\ if\ 2\le x\le3\\ 1,\ if\ x>3 \end{cases}$$ Find the value of ...
0
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1answer
33 views

sigma algebra of a stopping time

Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration. $\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma ...
1
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1answer
17 views

Finding bivariate probability mass function (by counting?)

Suppose that we role $d$ dice. Let $X, Y$ be random variables, where $X = \#$ rolled by the die with the highest value. $Y = \#$ rolled by the die with the second highest value. By convention, we ...
2
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2answers
20 views

Prove: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$

I'm trying to prove the following theorem using the axioms quoted below. Theorem: If $A \subset B$ then $P(A) \le P(B)$ and $P(B-A) = P(B)-P(A)$ Axiom 1: For every event $A$ in the class ...
2
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0answers
16 views

Convergence of sum dependent random variables in $L^2$ with mean zero

Suppose that $X_1, \ldots$ were a sequence of random variables with $E(X_i)=0$ and $E(X_i^2)= \sigma_i^2$ and that $S_n := X_1 + \cdots + X_n$ converged in $L^2$. Does this imply $\sum \sigma_i^2 < ...
2
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0answers
20 views

A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
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0answers
21 views

Probability density function above a given value. $\{ f(x) > c\}$

Say $X$ is a stochastic variable with a distribution $\nu$ and $f$ is the corresponding Lebesgue-measurable density. If I want to calculate a set $$A = \{ x \in \mathbb{R} \ | \ f(x) > c \}$$ for ...
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1answer
28 views

A doubt on a proof of a theorem of Durret's Probability Theory

Below is the text of the theorem: $\mathcal{F}_{i,j}$ are sigma algebras indexed by $i$ and $j$. I'm having some difficulties in understanding this proof. Do the $\mathcal{A}_i$ contain $\Omega$ ...
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0answers
23 views

What is a Tail Field and how to interpret it?

I cannot understand or form a good intuition in my head of what a tail field is. An introduction to rigorous probability theory by Rosenthal gives the following definition: Given a sequence of events ...
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0answers
47 views

Hypergeometric distribution with a priori probabilities of the balls

If we have a urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...
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0answers
9 views

How to calculate mean of busy-cylce

Let $\{X(t)\}$ be birth-death process on a finite state {0,1,2} with non negative birth rates $(\lambda_0,\lambda_1)$and death rates $(\mu_1,\mu_2)$. Suppose $\mathbb{P}(X(0)=0)=1$ and $s_0=\inf ...
2
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1answer
34 views

Disentangling $\int_Af(\mathbf{x})\ d\mathbf{x}$, using Fubini Theorem.

Let $\mathcal{B}^n$ be the borel sigma algebra generated by the rectangles in $\mathbb{R}^n$. I can write $f(\mathbf{x})=g_1(x_1)\cdots g_n(x_n)$. Let $\mu=\mu_1\times \cdots \times \mu_n$ be the ...
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0answers
15 views

Integrals of functions of statistics

Let $X: \Omega \to \mathbb{R}^n$ be a measurable random vector with law $\Lambda_X$ and probability density function (pdf) $f_X$. Let $T:\mathbb{R}^n \to \mathbb{R}^2$ a statistic (a ...
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2answers
27 views

Prove that if $X_n \xrightarrow{P} c$, then $E(X_n) \to c$ for $X_n$ uniformly bounded

I have been trying to prove that for a random variable that is uniformly bounded, i.e. $|X_n - c| <M$ for all $n$, convergence in probability to $c$ implies that $$E\left(X_n \right) \to c$$ ...
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0answers
8 views

How to find expectation of birth-death process

Let $\{X(t)\}$ be birth-death process on two-state space {0,1}. Let birth rate $\lambda=2$ and death rate $\mu=12$. How to calculate $\lim_{t\to\infty} \mathbb{E}[X(t)]$?
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20 views

What pdf or df corresponds to this mgf? [duplicate]

Suppose we have random variables $X_1, X_2, ...$ in $(\Omega, \mathscr F, \mathbb P)$ s.t. $P(X_n = k) = \frac 1 {n+1}$ for $k = 0, 1, ..., n$ and $X := \lim \frac{X_n + 1}{n+2}$ exists. ...
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1answer
30 views

There is problem in calculating pgf(probability generating function)

I posted question about distribution of poisson distribution multiplied by constant. Here! From this post, i can obtain what i want. $$P(X=x)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$ $$Z=\alpha X ...
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1answer
26 views

What is distribution of Poisson multiplied by positive constant

Let $X$ is poisson distribution. $$f_{X}(n;\lambda)=\frac{\lambda^{n}}{n!}e^{-\lambda}$$ And there is some positive constant $\alpha$. I like to know pmf(probability mass function) of $Z=\alpha X$. ...
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1answer
24 views

Calculate the estimators of $E[X]$ and $Var[X]$ using the method of moments

$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$ Demonstrate ...
2
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0answers
35 views

Is it possible to upper bound conditional expectation expression: $\mathbb E[X | X > c] - c$?

As the titles suggests, I am trying to see if we can upper bound $\mathbb{E}[X \text{ }| \text{ }X > c] - c$ For now, I am assuming bounded mean on both sides: $0 < m \leq \mathbb{E}[X] \leq ...
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1answer
13 views

How to calculate the values of a version of a conditional expectation

So I am just learning about conditional expectation in the modern probability sense. I understand that it is a random variable. What I am having a hard time understanding is how you calculate the ...
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0answers
32 views

Is the Covariance of Two Random Variables Convex or Concave or Neither?

Are there any standard results established regarding the behavior of the Covariance of two random variables? For example, whether it is a convex or concave functions and so on and under what ...
2
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0answers
29 views

Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
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0answers
8 views

Algebra generated by point cylinders

Let $X\equiv\mathbb N^{\mathbb N}$ denote the set of all sequences of positive integers. For a fixed $n\in\mathbb N$ and $(y_1,\ldots,y_n)\in\mathbb N^n$, define the “point cylinder” as follows: ...
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0answers
22 views

Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
2
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2answers
16 views

Sum of the union of probabilities proof

I am unsure how to prove that $P(A_1 \cup \cdots \cup A_n) \leq \sum_{k=1}^n P(A_k)$. The left hand side looks like the the general inclusion-exclusion formula, but the right side does not. However, ...
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1answer
29 views

Alternate proof to Rosenthal (3.6.5)

Rosenthal (3.6.5) Let $(\Omega, \mathcal{F}, P)$ be a probability triple such that $\Omega$ be countable, and $\mathcal{F}=2^{\Omega}$. Prove that it is impossible for there to exist a sequence ...
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2answers
54 views

Different solution of probability problem from textbook

It is the problem 1.2.3 of Karlin's book An introduction to stochastic modeling: A population having $N$ distinct elements is sampled with replacement. Because of repetitions a sample of size r ...