Questions tagged [sigma-algebra]
Sigma-algebras (or $\sigma$-fields or $\sigma$-algebras) define which subsets of the sample space $\Omega$ are considered events for the purposes of computing probabilities. Sigma-algebras are often denoted $\mathscr{F}$. They are used in the definition of a probability space which is a triple $(\Omega, \mathscr{F}, \mathbb{P})$ which precisely defines how probabilities will be computed.
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what is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure? [closed]
The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $...
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Sigma field generated by an arbitrary random variable
Q) Take $\Omega=[0,1]$, with the $\sigma$-field of Borel sets and $P$ the Lebesgue measure on $[0,1].$Find $E(\xi |\eta)$ for $\xi(x)=2x^2$,
$\eta(x) = \begin{cases} 2, & \mbox{if } x\in[0,\frac{1}...
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What's the relationship between these two definitions of martingales?
On wikipedia, the definition of a martingale is given as follows:
A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) $X_1, ...
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Finite additive probability defined on a "finite-additive" field
Countably additive probability is defined on sigma field. However a finite additive probability needs only a "finite-additive" field: the finite additive probability does not need the ...
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Plot indicating Tower Property of conditional expectation
During my work I encountered a plot, in which we have two curves indicating:
Mean value of observations of dependent variable $Y$ (in my case it was annual frequency of claims), calculated for ...
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Prove that smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$ [duplicate]
Let $A$ be a Borel set of $\mathbb{R}$. Then show that the smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$.
I am ...
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Prove that smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$
Let $A$ be a Borel set of $\mathbb{R}$. Then show that the smallest $\sigma$-field of subsets of $A$ containing the open sets in $A$ is $\{B \in \mathcal{B}(\mathbb{R}) \mid B \subseteq A\}$.
I am ...
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Intuition about conditional probability given a $\sigma$-algebra [duplicate]
I've been studying some statistics by the ways of measure theory, and came up with a problem in understanding conditional probability. The book gives the following definitions:
Let $(\Omega,\mathcal{...
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Actual definition of $\sigma$ algebra generated by information available up to time t-1 in ARCH, GARCH models
I'm trying to figure out the proper mathematical definition in sigma-algebra generated by infinite past in ARCH, GARCH type models.
Let $\epsilon_t = \sqrt{h_t} z_t $, where $z_t$ $i.i.d$ random ...
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Calculating μ(A\ B) in probability space
I am new in learning sigma-algebra. My question is related to computing μ(A\ B). I just need the hints and ideas on how to do it, I will do it.
Let (X,Ao,μ) be a probability space. It is given that A, ...
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What is the probability that multiple spectral lines are just noise?
In spectroscopy spectral line emissions are related to each other, with fixed widths and heights that correlate with all lines based on fixed physical conditions. If I detect one spectral line at a 3 ...
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How do we choose the sigma algebra for a specific experiment?
I am trying to understand the connect between probability and measure theory. I get some basic things about measure theory. But what I still don't get is how do we choose the sigma algebra for our ...
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Where does the random come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$?
For continuous random variables $X, Y$ the conditional expectation $\mathbb{E}[X | Y]$ is itself a random variable. I understood this in the sense that for a realisation of $Y$ we can say
$$
\mathbb{E}...
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Quick short question about understanding of conditional expectation
let me just ask one simple question, I am not sure if I understand this concept of conditioning w.r.t. sub-$\sigma$-algebras.
Let $(\Omega,\mathcal{A},\mathbb{P})$ be probability space and $X,Y:\Omega\...
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Strict and Real example of Random variable and distribution [duplicate]
Now I'm studying elementary probability thoery.
And It changes every abstract notions to strict text.
I mean, when I was in middle school, the probability of two times coin toss is like below.
$$P(HH) ...
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Independence in terms of sigma fields
I was solving a homework problem that asked for an example of uncorrelated but dependent random variables.
I'm using the example $X$ is uniformly distributed on $[-1,1]$, and $Y=X^2$.
It's ...
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Determine a product's rating given a known 3-sigma tolerance
Consider a 100Ω resistor with a 10% tolerance. We can assume this is the 3-sigma value since this is typical in manufacturing. Thus, we can expect ≈99.73% of such resistors to range from 90Ω to 110Ω.
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How to determine sample space, $\sigma$-algebra and probability measure from the exponential family?
The sample space of binomial distribution is the set $\{0,1\}$ and its $\sigma$-algebra is the power set of $\{0,1\}$ while the sample space of normal distribution is $\mathbb R$ and its $\sigma$-...
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Why is the probability measure P not defined on the power set of $\Omega$, but on it's sigma algebra? [duplicate]
why do we not define P on the power set of $\Omega$, which is equivalent to declaring that all sets are measurable. Even if $\Omega$ is uncountable, it's sigma-algebra and power set would also be- ...
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Is there a sigma-algebra for the parameter space in relation to the likelihood function?
The likelihood function, $L(\theta|x) = f(x|\theta)$, is sometimes mistaken to be a pdf. I have always thought that showing an example where
$ \int_{-\infty}^{\infty} L(\theta|x) d \theta = 1 $ does ...
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Do the following equations for the conditional expectations hold under the given assumptions?
Let $(\Omega,F,\mathbb{F}=(F_n)_{n\in\mathbb{N_0}},P)$ be a filtered probability space and $(Z_n)_{n\in\mathbb{N_0}}$ be a sequence of integrable iid random variables. Denote $F_n=\sigma(Z_o,\dots,Z_n)...
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How to understand conditional expectation w.r.t sigma-algebra: is the conditional expectation unique in this example?
This board has many questions on how to understand conditional expectations w.r.t a $\sigma$-algebra; it's clearly a topic that confuses many. I am one of them. I read all the other questions and ...
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Simple example of the $\sigma$-field generated by a random variable (Concept check)
$\Omega = \{ {\omega_1, \omega_2,\omega_3} \}$ where each state is equally probable.
Two random variables exist $\widetilde{x}$ and $\widetilde{y}$ that are functions of these states:
$\widetilde{x}(\...
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How to intuitively visualize measure theory concepts via diagrams? [closed]
Visualizing Measure Theory via Diagrams / Drawings / Geometry
Hi,
I'm starting to learn about Measure Theory in order to study more advanced topics in Data Science. However, the explanations are ...
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Sigma algebra generated by random variable on a set with generators
I'm having trouble proving an intuitive result I found in these lecture notes I'm using for self-study (1.2.14 there).
Suppose $X$ is a $(\mathbb{S}, \mathcal{S})$-valued random variable (from $(\...
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Does product sigma algebra of n B(R) (Borel) coincide with B(R^n)?
My question comes from different modes of probability law, which I met. I see two options in equality: set belonged to $B(\mathbb R^n)$ and Cartesian product of B belonged to $B(\mathbb R)$.
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Do likelihood functions require sigma-algebras as probability spaces do?
In statistics, the likelihood function (often simply called the likelihood) is formed from the joint probability of a sample of data given a set of model parameter values; it is viewed and used as a ...
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Why are p-values probabilities rather than likelihoods?
The p-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct.
Why is the p-value a ...
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covariance in 2 dimensional field
I want to know is the covariance equation which is shown below is how valid when we are in 2-dimensional fields (not 2-dimensional data)?
$$Cov(x,y)=\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-y)}{N-1}$$
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Independence of a Gaussian random variable and the product of another Gaussian random variable and a Bernoulli random variable
Let $X$ and $Y$ be two independent Gaussian random variables with mean $0$ and variance $σ^2_X$ and $σ^2_Y$ respectively. Let $Z$ be a random variable measurable with respect to $σ(Y)$ and suppose ...
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Why must probability fields be closed under countable unions?
Assume a probability triplet $(\Omega, \mathcal{F}, \mathbb{P})$.
My current understanding of $\mathcal{F}$ is that it must define events i.e. the subsets of $\Omega$ where probability is defined. I ...
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Difference between tight and uniformly tight random variables?
This wikipedia page implicitly says that “tight” and “uniformly tight” random variables refers to the same concept.
I find this somewhat surprising. Are there contexts in which a distinction is made ...
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Help with $\sigma$-algebra probability theory concept [closed]
Let $\Omega = \{0,1\}^{\mathbb{N}} = \{\alpha=(\alpha_1,\alpha_2,...):\alpha_i \in \{0,1\}\}$
Fact. There exists a $\sigma$-algebra $\mathcal{F}$ such that for every $\beta = (\beta_1,...,\beta_n)\in\{...
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Understand conditional expectation w.r.t. sigma -algebra [duplicate]
When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is
$$
E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
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What does the meet of two sigma algebras mean?
I came across this notation of which I am unfamiliar;
$\mathscr{F}=\mathscr{G}_{1}\vee \mathscr{G}_{1}$
where $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$ are both sigma-fields of subsets of $\Omega$. It ...
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Why is a random variable being a deterministic function of another random variable mean that it is in the sigma algebra of the other variables?
Specifically, I'm learning about martingales in class right now.
Given random variables $T$, $X_1, X_2, \ldots, X_n$, textbook that I'm reading draws an equivalence between the statement that the ...
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Can a sigma field ever be same as an event space?
Suppose, The sample space of the gender of the upcoming baby is:
$S=\{boy,girl\}$
then, $\sigma -field=\{ \{boy\}, \{girl\}, \{boy,girl\}, \emptyset\ \}$
Can the event of having the baby as twins ...
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Atoms of a sigma algebra
I've been reading Schilling's Measures, Integrals, and Martingales, and I ran across a remark (on page 21) that I don't understand.
Here is the setup: let $A_{1},\ldots,A_{n}$ be non-empty, ...
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What is the complement of an event in a sigma algebra?
Consider $\Omega=\{1,2,...,6\}$ and $F=\left\{\{1\},\{3\},\{5\}\right\}$. Let $A={1}$. If $F$ is to be a sigma algebra, then necessarily $A^c \in F$. What is $A^c$?
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Is an event a subspace of the sample space?
In a lecture today, a professor of mine described an event as being "in" the sample space. When writing on the board, for a sample space $S$ and event $E$, it was denoted: $$E \in S $$
This confused ...
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Understanding Borel Sets in Relation to Distributions
Following up with my last question, I am self studying the book Elementary Stochastic Calculus with Finance in View. The author has lost me in some of the terseness of his explanation of distributions....
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A possible typo in the textbook?
On page 74 of Lehmann's Testing Statistical Hypothesis, the author writes
Let $P_0$ and $P_1$ be probability distributions possessing densities $p_0$ and $p_1$ respectively w.r.t. a measure $\mu$
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Is there a reason other than conventions why a CDF must be defined for all real numbers
There are many cases when the sample space is not the entire set of real numbers (for instance a Bernoulli trial or sampling from an interval).
On the one hand: for the definition $F_X(x) = P(X \leq ...
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$\int_{G}E[X|\mathcal{G}]dP = \int_{G}XdP$ implies that $E[X|\mathcal{G}]=X$
I need to prove that for $\mathcal{G} = \mathcal{F}$ (so the largest sigma field), $\int_{G}E[X|\mathcal{G}]dP = \int_{G}XdP\, \forall G \in \mathcal{G}$ implies that $E[X|\mathcal{F}] = X$ almost ...
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What is it meant with the $\sigma$-algebra generated by a random variable?
Often, in the course of my (self-)study of statistics, I've met the terminology "$\sigma$-algebra generated by a random variable". I don't understand the definition on Wikipedia, but most importantly ...
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Find $\sigma$-field $\sigma(\mathscr A)$ generated by $\mathscr A$
Let $\Omega =\{1,2,3,4\}$, and let $\mathscr A = \{\{1\},\{2\}\}$.
Find $\sigma$-field $\sigma(\mathscr A)$ generated by $\mathscr A$.
My answer is $\sigma(\mathscr A) = \mathscr A \cup \mathscr A^c =...
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Computing the frequency into which x falls into the Borel $\delta$-algebra
I'm currently trying to make sense of the papers:
Clustering processes by Daniil Ryabko (link)
Online Clustering of Processes by Azadeh Khaleghi (link)
Both of them make use of the following metric
...
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How to work with an asymmetrical distribution
I'm looking at estimated and actual time taken for a range of projects, as these vary in length quite a lot I've normalised them so they are just estimate/actual, so an estimate of 16 days work that ...
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Exact meaning of conditional expectation $\mathbb{E}[X|\mathcal{F}]$
I'm going through elementary literature on measure theory from Shreve (Vol II) and having a hard interpreting the meaning of $\mathbb{E}[X|\mathcal{F(t)}]$ where $X$ is a random variable and $\mathcal{...
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A sigma field question
This is real basic, but a question that is annoying me:
$\Omega = \left \{ 1,2,3 \right \}$
$T = \left \{ \emptyset , \Omega , \left \{ 1,2 \right \}, \left \{ 2,3 \right \}, \left \{ 1,3 \right \}...