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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1answer
53 views

Markov chain doesn't sum up to 1

Let $\{X_n\}$ be a Markov chain on $S=\{1,2,3,4,5,6\}$ with the matrix suppose we define a new sequence $\{Y_n\}$ by $$Y_n=\cases{1\quad X_n=1\vee X_n=2\\2\quad X_n=3\vee X_n=4\\3\quad ...
3
votes
0answers
26 views

Existence of regular conditional distribution of random variable given the value of another variable

Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space with a measurable function $Y: (\Omega, \mathcal{A}) \rightarrow (E, \mathcal{E})$ and another measurable function $X: (\Omega, ...
0
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0answers
17 views

Have I used the Probability generating function of poisson point process correctly?

Let $v\in \mathcal{V}$ be measurable and let $\Phi$ be a Poisson Point Process with intensity $\lambda$ then the probability generating function (PGF) is $$\mathbb{E}\left( \prod_{x\in \Phi} ...
0
votes
1answer
15 views

Is the product measure space generated by the filtration adapted to the projection maps?

Let $(\Omega, \mathcal A)$ be a measure space. Consider the product measure space $(\Omega^{\mathbb N}, \mathcal A^{\mathbb N})$ and denote by $\pi_n : \Omega^{\mathbb N} \to \Omega$ the $n$-th ...
0
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1answer
41 views

Find the Expected value of a Random variable

Assume random variable $$X \sim f_X(x) \,\,\, -2 \leq x\leq 2$$ Now Assume we need to compute the following $$F= \mathbb{E}\left(\frac{1}{1+(G(X))^2}\right)$$ where we define the function $$G(x) = ...
0
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0answers
24 views

Show Projection minimizes variance

Van der Vaart's Asymptotic Statistics, problem 11.2 Another idea of projection is based on minimizing variance instead of second moment. Show that $\text{Var}[T-S]$ is minimized over a linear space ...
1
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0answers
22 views

Central limit theorem with Lyapunov condition

$Z_1, Z_2,...$ are iid uniformly distributed on $[-1;1]$, $\lim_{n \to \infty} a_n=0$ and $\lim_{n \to \infty} na_n=\infty$ also $a_n>0$ $\forall n$, $X_{n,j}= \frac{1}{a_n}I(|Z_j| \le a_n)$ ...
0
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1answer
20 views

Probability issue given a Bayesian Network

If we have a Bayesian Network A -> B ->C then P(B|A, C) = P(B|A)? Thanks!
0
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0answers
14 views

Proof of theorem on markov chains

If $X=(X_n)_{n\in\mathbb N}$ is a Markov chain on a space $E$, it has an initial distribution $(\lambda_i)_{i\in E}$ such that $\sum\lambda_i=1$ and a transition matrix $(p_{ij})_{i,j\in E}$ such that ...
1
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2answers
50 views

If $\{X_n\}$ converges in probability to $1$, where does $\{1/X_n\}$ converge to?

Without using the continuous mapping theorem, I want to show that, given $\{X_n\}$ is a sequence of random variables converging in probability to $1$, $\{1/X_n\}$ converges in probability to $1$. The ...
-1
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0answers
13 views

Chain conditional probability issue [closed]

In conditional probability network if A -> B ->C then P(B|A, C) = P(B|A)? If no, then what is the answer? Thanks!
2
votes
1answer
19 views

Slutsky for joint convergence

I am interested whether Slutsky's Theorem also holds in the case of joint convergence. Let $(X_n,Y_n)$ be random variables with $(X_n,Y_n) \rightarrow (X,Y)$ in distribution for $n \to \infty$. ...
0
votes
6answers
91 views

How come everyone says that you can't with in lottery because of statistics yet every single day I hear that someone has won?

I'm a very simple man with basic understanding of mathematics and theory. This question has bugged me for the last few years, ever since I learned about lottery tickets. When I talk with people about ...
0
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1answer
23 views

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network?

Which is the difference between $P(A \mid B)$ and $P(A=t \mid B)$ in a Bayesian Network, where $A$ and $B$ are boolean values?
-1
votes
0answers
24 views

Conditional probabilities given the evidence(Bayesian network)

Let's say we have a Bayesian network: How can I compute P(A | F, E) ? I have all the probabilities for each node. Thanks!
0
votes
0answers
24 views

Borel Sets and relation to probability theory.

I am currently having difficulty understanding the link between Borel Sets and Probability theory. How/Why are Borel Sets used in Probability theory?
1
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1answer
37 views

An Elementary Convergence Problem in Probability

Suppose that $X_1,X_2,...$ are degenerate random variables such that $f_{X_n}$ denotes the mass function of $X_n$.$$f_{X_n}(x)=P(X_n=x)= \begin{cases} 1, & x=2+\dfrac{1}{n} \\ 0, ...
2
votes
1answer
60 views

Independence of a Stochastic Process at Distinct Time

Suppose $X_t$ is a stochastic process of $t$ on $[0,\infty)$ with almost surely continuous sample path. I have modified my question to the following one, per Math1000's comment below: Is the ...
0
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0answers
26 views

Density of product of random variable

I would like to derive the product density of two independent continuous random variable in a measure theoretic framework. I am well aware of the result which can be found here: ...
2
votes
1answer
45 views

Counterexample for generating function?

This is Exercise 3.1.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Give an example for two different probability generating functions that coincide at countably ...
0
votes
0answers
17 views

Convolution of negative binomial distribution w/ generalized binomial theorem

This is Exercise 3.1.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Show that $b^−_{r,p} \ast b^−_{s,p} = b^−_{r+s,p}$ for $r, s \in (0,\infty)$ and $p \in (0,1]$. ...
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0answers
41 views

Origin of $\sigma$-algebra

In what paper, article or book was the notion of an $\sigma$-algebra first defined or mentioned? Or at least how far could this concept traced back?
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0answers
19 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
-1
votes
2answers
28 views

Convolution of 2 uniform random variables

I really do not know how to do this. Let $X$ have a uniform distribution on $(0,2)$ and let $Y$ be independent of $X$ with a uniform distribution over $(0,3)$. Determine the cumulative distribution ...
0
votes
1answer
35 views

If $F(a) - F(a^{-})$ is continous then $F(a)$ is continous [closed]

Suppose $F$ is a distribution function and, $$H(a) = F(a) - F(a^{-})$$ is continous for all $a \in \mathbb{R} $, where $$F(a^{-}) = \lim_{\epsilon \to 0^+}F(a-\epsilon)$$ How to prove that $F$ is ...
0
votes
0answers
10 views

The expected number of mutations in a sequence of elements, each with random delays

In a sequence, the number of the permutations, is the (minimum) number of the pair of elements needed to switch to make them sorted. For example in the following: ...
0
votes
0answers
14 views

Sigma field generated by the union of a field and a set

I am trying to show that; If $H$ is a set lying outside a field (or $\sigma$-field) $\mathcal{F}$, then the field (or $\sigma$-field) generated by $\{\mathcal{F}\cup\{H\}\}$ consists of sets of the ...
1
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0answers
13 views

Central limit theorem rate of convergences for different distributions

A famous fact in statistics is that for any i.i.d. random variables $X_1,\dots,X_n$ with mean $\mu$ and variance $\sigma^2$ $$\sqrt{n}\left(\frac1n \sum_iX_i - \mu\right)$$ approaches $N(0,\sigma^2)$ ...
3
votes
0answers
41 views

Calculation of Conditional Expectation $\Bbb E[f(X)\mid Y]$

$\newcommand{\Cov}{\operatorname{Cov}}$and thank you for taking the time to read this. :) My question is about evaluating $\Bbb E[f(X) \mid Y]$ (a random variable in $Y$). There's plenty online (and ...
0
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0answers
31 views

Size of families: Birth death immigration

The context of this problem is as follows. Starting from a population size of zero, immigrants arrive according to a homogeneous Poisson process with rate $\theta$. Once they arrive, immigrants start ...
1
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0answers
18 views

Comparing infinite dimensional distributions

Given two infinite sequences of rvs $(X_{1},X_{2},...)$ and $(Y_{1},Y_{2},...)$, how can we show $(X_{1},X_{2},...)\stackrel{d}{=}(Y_{1},Y_{2},...)$? The way I heard is by comparing all their finite ...
-1
votes
1answer
20 views

Show countable additivity of a certain probability measure

Let $\mathcal{F}$ be the field consisting of the finite and the co-finite sets in an infinite and ${\bf{uncountable}}\;\Omega$, and define a probability measure $P$ on $\mathcal{F}$ by taking $P(A)$ ...
0
votes
1answer
35 views

Statiscal Distance Properties

Please anyone could give me any idea of how prove the following property of statistical distance: $d(AB,CD)\leq d(A,C)+d(B,D)$ Remenber that: $(X,d)$---> Metric Space $d:X\times X\rightarrow ...
1
vote
2answers
52 views

If $X$ is a random variable with distribution $\mu$, prove $\int \limits_{\Omega} X(\omega) \, dP(\omega) = \int \limits_{\Bbb R} x \, \mu(dx)$.

I'm trying to prove $\int \limits_{\Omega} X(\omega) \, dP(\omega) = \int \limits_{\Bbb R} x \,\mu(dx)$ if $X$ is a random variable defined on $(\Omega, \mathcal{F}, P)$. I understand how to prove ...
0
votes
0answers
18 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
0
votes
0answers
15 views

Density of product of random variables

I am trying to calculate the product of two random variables, one that is exponentially distributed and the other that is uniformly between $[1, 2]$. Consider the following approach. We first ...
3
votes
2answers
87 views

Limit superior of $\sum_{j=1}^n X_j$ with $\mathbf{P}[X_j = 1] = \mathbf{P}[X_j = -1] = 0.5$

This is Exercise 2.3.1 from Achim Klenke: »Probability Theory — A Comprehensive Course«. Exercise: Let $(X_n)_{n\in N}$ be an independent family of $\mathrm{Rad}_{1/2}$ random variables (i.e., ...
0
votes
0answers
25 views

Entry time and hitting time

Hi I have a question about entry time and hitting time. Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_{t})_{t \in[0,\infty)}$ be a $\mathbb{R}$-valued stochastic process on $(\Omega, ...
1
vote
2answers
20 views

When does probability mass outside a sufficiently large ball is small?

Many times when I read books about statistics or probability theory, I encounter proofs which said: For any $\epsilon>0$ there is an $M\in(0,\infty)$ such that $\text{Pr}\{X\in ...
2
votes
1answer
62 views
+200

Correlation of a vector generated and its one-period lag, both generated using AR(1) data

Suppose that $C_0$ is $100$ and $\{e_t\}_{t\geq 1}$ is a sequence of i.i.d. standard normal random variables. We generate $C_t=C_{t-1}+e_t$ for $t\geq 1$ and set $$ x_t=C_t^2-C^2_{t-1},\quad ...
1
vote
0answers
20 views

How to find mean and variance for probability problem with warranty?

I am in a probability theory class and I'm stumped on a problem: A warranty is written on a product worth \$10,000 so that the buyer is given \$8000 if it fails in the first year, \$6000 if it fails ...
1
vote
0answers
11 views

Proof of Wald's Identity, is this valid?

So Wald says that assuming that $T$ a stopping time and $X_i$ i.i.d. variables are $L^1$, that $E[S_{T}] = E[T]E[X_1]$ given that $S_n = \sum_{i=1}^n X_i$. Consider the following proof that is ...
3
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0answers
32 views

Condition for $L^p$ convergence of backwards martingale

Is there any condition that is known to be sufficient for $L^p, 1<p<\infty$ convergence of a backwards martingale (and why is it sufficient)? I couldn't find anything else than the normal $L^1$ ...
0
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0answers
9 views

Cauchy distribution derivation

So now I'm doing a different example with the Cauchy distribution by letting $Z=\tan(U)$ for $U$ distributed between $[−π/2,π/2]$. So then $P(\tan(U)≤a)=P(U≤\arctan(a))$, which is equivalent to the ...
1
vote
1answer
41 views

How to approach analyzing probability problems? (Specific question included)

I've recently become very interested by the concept of probability. After doing some studying, I believe I've become fairly familiar with the terms: probability, random variables, probability ...
0
votes
0answers
15 views

Kullback-Leiber divergence for two simple probability vectors

For any probability vectors $ p= (p_1,...,p_K) $ and $ q=(q_1,...,q_K) $ representing monotonically increasing functions $ x-1 $ and $ ln(x) $ respectively what is the KL divergence? $$ KL(p||q) = ...
0
votes
1answer
18 views

Probability density function of function of a random variable [closed]

What is the probability density of say $e^{a + bX}$ if $X$ is normally distributed?
1
vote
1answer
15 views

Doob Decomposition is $L^1$ bounded

Suppose $X_n$ is a martingale that is $L^p$ bounded for some $p > 1$. Then the problem asks to show that the Doob Decomposition of the submartingale $|X_n|^p = M_n + A_n$ where $M_n$ is a ...
0
votes
1answer
62 views

Conditional expectation w.r.t. random variable and w.r.t. $\sigma$-algebra, equivalence

Let $\Omega = \{ \omega_i \}$ be a countable set, and consider some probability space $(\omega, \mathcal F, P)$ with $p_i := P(\{ w_i \})$. Let $X : \Omega \to \mathbb R$ be a random variable, then ...
0
votes
1answer
10 views

Convergence in distribution of a normalized Poisson distributed random variables

Show using the central limit theorem that $\frac{X_n-n}{n^{1/2}}\rightarrow Z$ where $Z$ is standard normally distributed and $X_n$ is $Poisson(n)$ distributed.