Questions tagged [notation]

For questions about statistical notation and mathematical notation used in statistics.

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Explanation of $X: (\Omega,\mathscr A,P)→(\mathbb R , \mathscr B',\mu)$ [duplicate]

A random variable $X$ is a function with a finite value so that: $X: (\Omega,\mathscr A,P)→(\mathbb R , \mathscr B',\mu)$ I've seen this line but I don't know how to understand it. Can you please ...
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Mistake in Casella & Berger on page 207?

Page 28: A note on notation: Random variables will always be denoted with uppercase letters and the realized values of the variable (or its range) will be denoted by the corresponding lowercase ...
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What does the (statistical) operator $\mathbb{D}$ usually mean? [closed]

Im reading and trying to understand the following paper on "Disentangled State Space Representations" (https://arxiv.org/pdf/1906.03255.pdf). In the derivation of the KL-loss term the ...
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Implied expressions with same probability

Given two variables $a$ and $b$ and two intervals $I$ and $J$, is the following affirmation true? $$ \Pr[a\in I]\geq p\ \land\ (a\in I\implies b\in J) \implies \Pr[b\in J]\geq p $$ Of course, $a\in I$ ...
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Is there an established way to write conditional probability for whole expression?

I am writing a methodological paper and I have expression like this, which is slightly more complicated bayes rule, because everything is conditioned by c: Is ...
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Hierarchical mixed-effects models notation

I have a hierarchical mixed-effects model where observations i, from group S_2 (representing municipalities), are nested inside group S_1 (representing states), which in turn are nested inside N (...
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conditional expectation of random vectors

For a random vector $X=(X_1, ..., X_n)^\intercal$ the expectation value can be written as $\mathbb{E}[X] = (\mathbb{E}[X_1], ..., \mathbb{E}[X_n])^\intercal$ according to equation 2 in https://en....
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What do "Expectation" mean in cost function?

I want to know whether my understanding for expectation of loss function is correct or not. (x,y) means sample data-label distribution from true distribution. And function f which is respect to ...
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Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it)

I have a problem with the concept of a symbol (an arrow with a P over it). Can anyone clarify this for me? For example what the symbol means in this relationship: $$\large Y_t^{(n)} \stackrel{p}\...
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Basic Notation Help

Taking a robotic planning class and have no stats background. I am just trying to understand what this means: "What is the probability of the event $R=\{x||x-1|\le 1\}$?" Can someone tell me ...
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Notation for Percentile Range of Sample

Let $X$ be a random variable with unknown CDF $F_X$ and PDF $f_X$. What is the notation for a percentile range $[0,p]$ on a sample of $X$ that allows discussing sample statistics over that range? My ...
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How to write the formulas for logistic & ordinal logistic regression models

In my thesis I am using logistic regression models (either binomial or ordinal, depending on the type of the dependent variable) to analyse the association between environmental quality and socio-...
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Is the expectation in the expression for Risk taken with repect to $\hat{\theta}$?

In Chapter 13 of Larry Wasserman's All of Statistics A Concise Course in Statistical Inference is given: Definition 13.1 The risk of an estimator $\widehat{\theta}$ is $$ R(\theta, \widehat{\theta})=\...
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Weird Term of Log likelihood

Recall the setup of logistic regression: We assume that the posterior probability is of the form $p(Y=1|x) = \frac{1}{1+e^{\beta^Tx}}$ This assumes that Y|X is a Bernoulli Random variable. We now turn ...
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What is the meaning of the tilde use in this probability expression? [duplicate]

$$X \sim \mathcal{N}(0,1)$$ $$Y|X = x \sim \mathcal{N}(x,1)$$ I'm familiar with the normal use of tilde in indicating a random variable follows a distribution (such as in the first line of the ...
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Regression Notation - does the distribution of $y_i$ refer to the distribution of $(Y|X)$?

The title question is the main focus of what I am trying to understand, but the reasoning behind my asking is more nuanced. I have recently made transition from learning about basic forms of ...
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Why is sample variance notated as $\frac{S_{x x}}{n-1}$?

I'm going through my class textbook and it defines sample variance as this: $$s^2 = \frac{\sum (x_i - \bar x)^2}{n-1} = \frac{S_{x x}}{n-1}$$ What exactly does $S_{x x}$ mean? Is it always synonymous ...
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Notation used in time series

I found this wikipedia article about the concept of stationarity in stochastic processes. Let $\left\{X_t\right\}$ be a stochastic process and let $F_{X}(x_{t_1 + \tau}, \ldots, x_{t_n + \tau})$ ...
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Likelihood notation - Bayesian models

I was reading up about variational Bayesian logistic regression here: https://www.inf.ed.ac.uk/teaching/courses/mlpr/2015/log/15b_lr_and_svi.pdf And the author first defines the model likelihood as $$ ...
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How do I write out the components of my beta regression, and how do I interpret them afterwards?

I am estimating a beta-regression using the logit-link function with a continuous dependent variable bound between 0 and 1 and 5 continuous predictor variables. In R the equation is given by: betareg(...
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Name of the formula which links an event with an alternatives' system

Given an event $B$ and a system of alternatives (I hope this is the right name in english, otherwise tell me the right one please) $(A_i)_{i=1}^n$, the following formula is valid: $$ P(B \, | \, \...
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Matrix algebra with notation question on $z_{.2}$ [closed]

I am doing some matrix calculations and came across an expression which I don't really understand: $$(z_{i2}-z_{.2})(z_{j2}-z_{.2})$$ As I understand it, those are two integers which are multiplied by ...
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How to show that $\mathbb E_{X,X'} \left[\sum_{i=1}^m\sum_{j=1}^mk(x_i,x_j) \right] = 2m\mathbb E_{x\sim p} [k(x,x)]$?

I posted a related question also to math.SE. Basically, I would like to know how to show the following, which is part of this paper: \begin{align} &\frac{1}{m} \mathbb E_{X,X'}\left[\sum_{i=1}^m\...
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Making predictions with support vector classification

I am reading this book (chapter 8.3 - 8.5) about SVM's, while also trying to work with the ksvm() function from the kernlab ...
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What does e+ mean in linear mixed model output? [duplicate]

In my mixed effect model output, the Estimate for Intercept is " 7.141e+00" I know e- but I am not sure what e+ is. Also, df has numbers like 5.461e+01, 1.052e+02, 5.396e+0. If that means ...
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1 answer
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Notation for the summation in unbalanced panel data

Unbalanced panel data, where $i$ stands for sections and $t$ stands for time points, have different numbers of $t$ for every $i$ (or different numbers of $i$ for every $t$, but let me use the first ...
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What is the difference between a realisation of a random variable and random variable itself?

I am having a hard time distinguishing random variables from their realisations. (Please note that for the sake of simplicity of my question I use discrete values in my example below.) Usually, a ...
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Is a tilde or an equals sign correct in linear mixed model formulas?

I know the formula for a linear mixed model (LMM) is often (always?) written with a tilde, rather than an equals sign, between the LHS and RHS. For example, one would write outcome ~ 1 + var1 + (1|...
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What does "$\pi$-almost" mean?

I'm reading "Complex Stochastic Systems" edited by Barndorff-Nielsen et al. I found that they use the expression $\pi$-almost consistently throughout Chapter 1. I don't understand what $\pi$-...
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Notation Clarification [duplicate]

Some functions (as an example) are written as $f(\theta;\mu,\sigma^2)$ and other occasions $f(\theta|\mu,\sigma^2)$ might be used. Can anyone clarify the difference in notation between using ';' and '|...
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1 answer
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Notation for an entire time series vs. one time instance

In time series analysis, $x_t$ may denote an entire time series (a time-ordered set of random variables or their realizations) or a single random variable or its realization that is specific to time $...
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What is the meaning of these subscripts in iid random variables? [duplicate]

I've been studying statistics on my own and I'm having a hard time understanding some notations. On this page: http://scipp.ucsc.edu/~haber/ph116C/iid.pdf, specifically on the second paragraph, the ...
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1 vote
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What is the meaning of the sum and the Sigma notation in the Goodfellow et al. simplified version of the back-propagation algorithm?

In the Goodfellow et al. book "Deep Learning" there is a simplified version of the back-propagation algorithm (Algorithm 6.2, Chapter 6). Authors state that the chain rule with respect to ...
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1 vote
1 answer
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using argmax in maximum a posteriori estimation from max likelihood estimation

I was going through maximum a posteriori estimation and could not understand, why argmax is used for first term in image shown below but missing from second term. Shouldn't we have used argmax for ...
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Wold Decomposition -- summation to infinity

In Wold's decomposition we have $ Y_t = \sum_{j=0}^\infty b_j \varepsilon_{t-j} + \eta_t $, where the variables have definition as in the Wikipedia page. I'm confused about why the summation goes to ...
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6 votes
2 answers
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What does a subscript on a probability represent?

I've been trying to look this up, but I must not be searching correctly because I think it's a pretty basic question. What I want to know is what do subscripts mean when talking about the probability (...
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5 votes
3 answers
143 views

What does $X=x|Y=y$ actually mean by itself?

I know that: $P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}$ But I'd like to understand what $X=x|Y=y$ means by itself, for example $(X=x,Y=y)$ means $X=x$ and $Y=y$. Would $X=x|Y=y$ mean? that $X=x$ when $Y=y$...
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Why isn't the Random Variable over which expectation is taken generally written down or noted as the argument of the expectation?

This is page 266 of the Causal Inference book by Rubin/Imbens. Note that the argument of the expectation is written down below $E_w$ but not in the first set of equations on the page. IMHO, it makes ...
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Question regarding binomial and bernoulli notation

I have a bunch of independent binary variables Xk where k >= 1 and P[Xk = x] = 1/4^u * 3/4^(1-u) for u = 0 or u = 1 ... this is a bernoulli distribution, right? I'm confused on whether I should ...
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1 vote
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Notation in Trust Region Policy Optimization by John Schulman et al

I am quite new to the area of reinforcement learning and find it hard to convice myself that the different notations used for reward function, state/action value function etc. coincide. Apparently I ...
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Abuse of notation : same function name for different distributions

Is this too much of an abuse of notation to use the same letter, e.g. $f$, to designate the joint - $f(x,y)$, marginal - $f(x), f(y)$, and conditional - $f(x|y), f(y|x)$ - probability/cumulative ...
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2 approaches for Monte-Carlo : weighted sum of $\chi^2$ distribution and Moschopoulos distribution with Gamma distribution

If I take as definition of $a_{lm}$ following a normal distribution with mean equal to zero and $C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})$, and if I have a sum of $\chi^2$, can I write the 2 ...
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1 vote
0 answers
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Mathematical Notation for Mixed Effect Model in Bayesian Framework

I would like to convert my bayesian mixed effect model into a mathematical notation and am unsure what the correct notation would be here. Can anyone help me or recommend a reference where I could ...
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2 votes
1 answer
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Notation for a derived probability measure

Suppose that I have a random variable $X$, which is a variable-dimensional (i.e. it could be 1 dimensional, 2 dimensional, 3 dimensional, etc.). The dimension of a specific $x$ is known and denoted as ...
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What is the mathematical notation in the attached figure means?

The full context can be found in this link: https://towardsdatascience.com/understanding-generative-adversarial-networks-gans-cd6e4651a29 Appreciate your kind help
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Multi-armed Bandits

Could someone explains to me the notation of this function, I mean I understand that we take the average of sum of the rewards for some particular action, however the notation seems strange to me for ...
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9 votes
1 answer
4k views

What is the difference between the dot product and the element-by-element multiplication?

What is the different between the dot product "$\cdot$" and the element-by-element multiplication notation $\odot$ in Statistics? I referred to Hamilton's Time-Series Analysis, and these ...
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4 votes
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Rao-Blackwellization in Black Box VI

In the paper, "Black Box Variational Inference," by Ranganath et al. (2013), the authors derive a Rao-Blackwellized estimator of the gradient of the evidence lower bound with respect to a ...
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2 votes
1 answer
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Different equations for linear mixed model

I don't have statistics background but I am studying LMM and trying to understand why different resources have different notation. Based on my reading I tried following as simple model with random ...
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1 vote
1 answer
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Notation in Bayesian hierachical models: what does * indicate [closed]

I am new to Bayesian Statistics and have a question about the notation *. What does it indicate in the context of hierarchical models ? Cheers
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