Notation refers to questions about statistical and mathematical notation.

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3
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23 views

Definition of $X_t$ in the context of Stochastic process and Time Series

In the book An Introduction to Stochastic Modeling , Stochastic process is defined as : A stochastic process is a family of random variable(s) , $X_t$ , where $t$ is a parameter running over a ...
2
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0answers
22 views

Control chart constants [on hold]

When calculating Upper Control Limit (UCL) and Lower Control Limit (LCL) in control charts I use constants from ...
2
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2answers
108 views

Which notation should be used for the mean, SD, and median: mean±SD or mean (SD)?

I usually use mean±SD notation for reporting values. I know that many people use mean (SD) notation and I understand why. Right now in the document I am writing I am also reporting medians but I ...
0
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1answer
25 views

Names for standard error equations

The equation for a standard error of the mean can be written like this: $\sqrt{\frac{s^2}{n}}$ or like this $\sqrt{\frac{\left(\frac{\sum (X - \bar{X}) ^2}{n-1}\right)}{n}}$ They both say the same ...
2
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0answers
26 views

Notation for a random vector whose length depends on another random variable?

I have the following process that I'm trying to describe with random variables. First, I have a random variable $X$ that takes on values drawn from a Poisson distribution with parameter ...
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0answers
34 views

Correctness proof of bayesian networks

I want something simple and my brain is getting in my way. Assume I have three different coins - C1 is fair, C2 has p(Heads)=0.6 and C3 has P(haeds)=0.8 I want to draw a bayes network for the ...
2
votes
1answer
52 views

Question about notation in Introduction to Statistical Learning

I've been working my way through the problems in the book "Introduction to Statistical Learning". I have a question about the notation in Question 5 from Chapter 3 (screenshot below). What does ...
2
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1answer
83 views

Bayesian linear regression question

I am doing a problem on Bayesian regression but I'm having a lot of trouble with it. Here is the question: Consider $f=w^Tx$, $p(w)\sim N(w|0,\Sigma)$. Show that $p(f|x)$ is Gaussian. Find the mean ...
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0answers
31 views

Question about notation of expectation operators over multiple random processes

My question is what are suitable or accepted notations for taking expectations over multiple random processes in the same equation. Let a model for variable $y$ be given by $$y_{i ...
-1
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1answer
36 views

What does posterior “over” parameters $\alpha$ exactly mean? [closed]

From my understanding the posterior "over" parameters $\alpha$ is $$p(D|\alpha)$$ and not $$p(\alpha|D),$$ is it correct?
1
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0answers
45 views

How can the output from a Cox proportional hazards model be mathematically translated into a log regression model? [closed]

I would like to write down the log-regression model associated with the formula: coxph(Surv(time, status) ~ rx + cluster(litter), rats) I am not interested in ...
1
vote
2answers
121 views

What does $\{x|x = 7\} = \emptyset$ mean? (Probability)

To my understanding $\{x|x = 7\} = \emptyset$ means that the number seven is a not allowed value. But I do not understand the meaning of "$x|x$". Can anybody please explain $\{x|x = 7\} = \emptyset$ ...
5
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1answer
127 views

Understanding the error term

I am trying to figure out the meaning of these different "hatted" terms in regression analysis. Here is my basic understanding: $Y$ the original of population/sample values $\hat{Y}$ regression ...
3
votes
1answer
47 views

Posterior of a simple Bayes linear regression

In the context of simple linear Bayesian regression, why or when is it appropriate to define the posterior as $p(\beta, \sigma^2|y)$ and not $p(\beta, \mu|y)$?
2
votes
0answers
25 views

Is it a well-established convention that pdf is lower case but CDF is upper case?

I'm wondering if the Wikipedia page for "probability density function" is wrong in having the acronym be all lower-case "pdf". The page for "cumulative distribution function" has all upper case "CDF". ...
1
vote
1answer
43 views

Why do some parameters not appear in the matrix equation for an ANOVA model?

Here is the question I have been working on and this is the answer of it I realised $\beta_3$ is not in the $\beta$ matrix. I don't understand why. Can anyone please explain this?
1
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1answer
49 views

Does a Bayesian network include the CPTs?

I'm preparing slides for a lecture, and I require some guidance. I'm only talking about discrete variables. How would you formally define the concepts surrounding Bayesian networks? A Bayesian ...
0
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0answers
31 views

$L_p$ norms for vector valued function in econometrics

What is meant in econometrics by $\|\mathbb{y}\|_p$, where $y$ is an $d$-dimensional random variable? Namely, is it true that $$\|\mathbb{y}\|_p = (E\|y\|^p)^{1/p},$$ where $\| \cdot \|$ denotes the ...
6
votes
1answer
168 views

What is the meaning of $\oplus$ and $\otimes$?

I am struggling to fully understand some notation in a book where they use a "crosshair" symbol - first like $\bigoplus\limits_{i=1}^n{} Z_j $ where the $Z_j$ are matrices and second like $I_n \otimes ...
2
votes
0answers
32 views

Standard error notation

Dealing with a standard error of a mean $$ \widehat{SE}_{\bar x} = \hat \sigma_{\bar x} = \frac{s_x}{\sqrt{n}}$$ $$ SE_{\bar x} = \sigma_{\bar x} = \frac{\sigma_x}{\sqrt{n}}$$ Is this the ...
2
votes
1answer
32 views

Symbol to indicate normalized or standardized variables

Is there a symbol to indicate that variables have been standardized? For example, if I have 2 different scoring functions Score1 and Score2. Let's say I want to form a combo score and show that the ...
8
votes
4answers
270 views

What is the difference between $E(X|Y)$ and $E(X|Y=y)$?

Generally, What is difference between $E(X|Y)$ and $E(X|Y=y)$? Former is function of $y$ and latter is function of $x$? It's so confusing..
5
votes
2answers
86 views

What is the correct notation for stating that random variables X and Y are independent?

Often people put down $X\perp\ Y$ as independence, but this merely means that the expectation of X and Y is zero and does not have any implication on correspondence between their joint PDF, CDF and ...
1
vote
1answer
95 views

Random Forest proof notation

I am having a bit of difficulty understanding the notation in equation (1) on page 4 of the following paper: https://escholarship.org/uc/item/35x3v9t4#page-4 Specifically, what do $E_{X,Y}$ and ...
2
votes
1answer
124 views

Best statistical notation for expected probability density

Assume that we have two multivariate normal distributions $\mathcal{N}_1 = \mathcal{N}(\mu_1, \Sigma_1)$ and $\mathcal{N}_2 = \mathcal{N}(\mu_2, \Sigma_2)$. We do these two steps: Pick a point, say ...
4
votes
3answers
77 views

What is the correct notation if two random variables belong to the same distribution?

I want to explain that both the real and imaginary part of a complex variable follow a zero-mean complex Gaussian distribution. How can I write that? For one variable I'd write $a \sim ...
0
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0answers
95 views

Did I write down my model correctly using matrix algebra and LaTex?

Model is essentially a seemingly unrelated regression estimation of the two following equations: ...
3
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2answers
149 views

What does the notation $(\textbf{X} \perp \textbf{Y} , \textbf{W}\mid \textbf{Z})$ mean?

I was reading Koller's and Friedman's Probabilistic Graphical Models book and became confused about some of its notation because of a set of notes that either contradict it or express it differently. ...
3
votes
1answer
264 views

What do vertical bars mean in statistical distributions?

What do the vertical bars mean in the first and third formulae? $$v_i|z_i=k,\mu_k\sim\mathcal{N}(\mu_k, \sigma^2)$$ $$P(z_i=k)=\pi_k$$ $$\pi|\alpha\sim \text{Dir}(\alpha/K1_K)$$ $$\mu_k\sim ...
3
votes
1answer
59 views

Notation: Deterministic Variable, Random Variable, Realization of Random Variable, Function

Is there a standard notation for distinguishing between the following: Random Variables Realizations of Random Variables Deterministic Variables (not random) Functions I am familiar with using ...
3
votes
2answers
32 views

X vs x Notation in law of large numbers

This may be a silly question, but I can't find a concise answer. I've been studying Convergence of Random Variables in Wasserman's All of Statistics, which starts out by explaining: $X_n$ is a ...
6
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2answers
106 views

Which notation and why: $\text{P}()$, $\Pr()$, $\text{Prob}()$, or $\mathbb{P}()$

Are these merely stylistic conventions, or are there substantive differences in the meanings of these notations? Are there other notations meaning "the probability of" that should be considered in ...
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2answers
55 views

Mixed variable, joint distribution, How do we know which one is continuous distribution, which one is discrete

If we have one continuous r.v. $x$ and a discrete r.v. $y$ which takes one of the two values $y_1$ and $y_2$. Let's say we know the prior probabilities $P(y_1)$ and $P(y_2)$. From Bayes theorem we ...
3
votes
1answer
104 views

Squared random variable $X^2$ vs $X\times X$

As I understand a random variable represents all possible outcomes of an experiment with their associated probabilities. Why $X^2$ is understood as squaring outcomes of experiments instead of as ...
0
votes
1answer
51 views

Correct notation wrt. uniform distribution

Assume that I have a discrete set $L$ and a transformation $\phi: L \rightarrow [0,1]$ that normalizes set $L$ such that now values belonging $L$ are uniformly distributed among the unit interval. ...
2
votes
1answer
66 views

Random variable as a function

I'm reading Schaum's outline of probability, random var. and random processes. In the second chapter they make it clear that a random variable $X$ is not a variable in the traditional sense but ...
1
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1answer
63 views

Correct way to write values which have a certainty of 95 percent?

Instead of perusing through many more textbooks, I thought it's easier to ask here. Say I would like to express the mean for a variable with a certainty of 95 percent, how exactly do ("should") I do ...
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0answers
27 views

What's the proper notation for a formula that includes values sampled from a distribution?

I'm trying to write an equation for a time series $\text{TS}$ containing noisy and unevenly sampled data from a function $m(t)$. Here's what I've come up with: $$\text{TS}(t \in \mathcal{U}(0, 1)) = ...
3
votes
1answer
83 views

dG(y) in expected value integral

I am wondering what exactly the notation dG(y) inside an integral means, what it's called and where I can read more about it: $$E[B_1]=\int_0^{\infty}E[B_1 \vert Y_1 = y] dG(y)$$
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0answers
30 views

How to simplify equation without using backshift notation and with “Yt” expression alone on left side?

I have three equations and don't understand why I missed points. All that was said was "simplify." B^4Yt = c + et (B-0.6B^3)Yt=(2+5B+0.8B^2)et Yt=(1+B^3/B)et
2
votes
1answer
140 views

What does the | in |Z| mean in mathematical expressions for distribution statistics

I am doing some self study on statistics and noticed that in the notes that I was using the $|Z|$ expression as attached in the photos below. I am confused with the "$|$" that is being used. The only ...
2
votes
3answers
99 views

Can I write estimate $\pm$ its standard error?

Suppose I have an MLE estimate $\hat{\theta}$ for a parameter $\theta$, and $\hat{\sigma}$ is the sqrt of the inverse of the negative of the Hessian of the log likelihood at $\hat{\theta}$. Can I ...
1
vote
1answer
179 views

Variance and covariance notation: $\sigma^2 V_1$, $\sigma^2 V_2$

I am reading an internal paper that says: Let $\sigma^2 V_1$ equal the variance of $\sum_{m\in M}Z_m - Z_0$ and $\sigma^2V_2$ equal the covariance of $||M||^{-1}\sum_{m\in M}Z_m - Z_0$ and $Z_m ...
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0answers
24 views

Angled brackets in notation (expectation or inner product doesn't seem to make sense)

I am reading an internal paper at my university that refers to volatility (for a variable not explicitly mentioned) as: $\langle X,X\rangle = \int_0^T \sigma_t \, \mathrm{d}t.$ I am unfamiliar with ...
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0answers
16 views

Question on Matrix Operation Notation

I have confused myself in attempt to express a 2-step operation in matrix notation-- The data consists of 2 "chunks" of information--quantity of identical items sold in different stores and their ...
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3answers
692 views

Is the “test statistic” a value or a random variable?

I am a student taking my first Statistics course now. I am confused by the term "test statistic". In the following (I saw this in some textbooks), $t$ seems to be a specific value calculated from a ...
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1answer
153 views

Logit link notation

Here is the whole question: (a)A Simulation Study (Logistic Regression). Assume $y|x \sim \mathrm{Binary}(p)$, where $p= \mathrm{E}(y|x)$, and $\mathrm{logit}(p_i)=-1+5.1x_{1i}-0.3*x_{2i}$. ...
8
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4answers
2k views

Is it correct to use 'Ln' instead of 'ln' for natural logarithm?

In some research papers, authors use 'Ln' for natural logarithm instead of 'ln'. Is it correct?
10
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4answers
415 views

Regression residual distribution assumptions

Why is it necessary to place the distributional assumption on the errors, i.e. $y_i = X\beta + \epsilon_{i}$, with $\epsilon_{i} \sim \mathcal{N}(0,\sigma^{2})$. Why not write $y_i = X\beta + ...
4
votes
1answer
164 views

Help Translating R GLM Command to Math Notation

I have the following generalized linear model. Object glmDV is modeled as a proportion of successes over total trials. Objects ...