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# What is the formalism that allows Random Variables to be treated algebraically like real or complex numbers?

We all know that if we have a variable x, then there is a meaning to - for example - $$y=e^x$$.

And we all know how to manipulate that algebraically and to do calculus. For example, if

$$y_1=e^{x_1}$$

and

$$y_2=e^{x_2}$$

then

$$y_1y_2 = e^{x_1+x_2}$$

But random variables are a very different beast. Ultimately we all know intuitively what we mean when we draw a PDF or, more accurately, the CDF which is itself an algebraic function mapping $$[-\infty, \infty]$$ to $$[0,1]$$ as part of that defines a random variable.

But what is the actual math that makes that definition of an RV actually act algebraically, so that, as above, we can happily write things like

$$Y_1=e^{X_1}$$

and

$$Y_2=e^{X_2}$$

then

$$Y_1Y_2 = e^{X_1+X_2}$$

where $$X_1$$ and $$X_2$$ are random variables - and get away with it???

• If you are worried about measurability issues, there are basic theorems that say the product of two measurable functions is measurable, and so on. Otherwise, the fact $Y_1(\omega)Y_2(\omega)=e^{X_1(\omega)+X_2(\omega)}$ for all $\omega$ follows by substituting the definitions of $Y_i(\omega)$ (as the answers below indicate) and the PDF/CDF has nothing to do with it...this is a more basic fact about substitution. – Michael May 22 at 2:22

Formally $$X_i$$ are real valued (measurable) functions i.e $$X_i\colon \Omega\to \mathbb{R}$$ for $$i=1,2$$ where $$\Omega$$ is the sample space. So $$Y_i=e^{X_i}$$ for $$i=1,2$$ as an equality of functions means that $$Y_i(\omega)=e^{X_i(\omega)}$$ for all $$\omega\in \Omega$$. Hence since $$X_i(\omega)$$ is a real number it follows using the first part of your question that $$Y_1(\omega)Y_2(\omega)=e^{X_1(\omega)}e^{X_2(\omega)}=e^{X_1(\omega)+X_2(\omega)}$$ for all $$\omega$$ whence as functions $$Y_1Y_2=e^{X_1+X_2}.$$
• It’s probably worth pointing out that one can also consider equality in distribution, i.e. $Y$ is a random variable with distribution equal to the distribution of $e^X$, as well as almost sure equality. – icurays1 May 22 at 0:26
• @eSurfsnake: Random variables need not take values in the interval $[0,1]$, the values can be anything. You might be confusing it with probabilites, which must of course be in the interval $[0,1]$. An event is a (measurable) subset of $\Omega$, and the probability of an event is its measure, which is at most $1$, since a probability space $\Omega$ by definition has measure $1$. – Hans Lundmark May 22 at 5:57
• yes, to me an RV is tightly bound with a probability - otherwise, what is the point of probability theory? But, am I right to assume that the abstract concept of a measurable function means precisely that if I have and RV, X, with its CDF, then a "measurable function" g(X) maintains certain logical relations; i.e., that a domain of measure $\mu$ on X maps to a range of meaningful measure such that no contradictions occur between X and g(x)? – eSurfsnake May 23 at 6:04
Formally a random variable is simply a measurable function $$X:\Omega\to \mathbb R$$, where $$\Omega$$ is a sample space equipped with some sigma algebra, $$\Sigma$$, containing the events we are interested in, and a probability measure $$\mathbb P$$. We call the triple $$(\Omega, \Sigma, \mathbb P)$$ a probability space. Thus if $$g:\mathbb R\to \mathbb R$$ is some "regular" function, $$g\circ X:\Omega\to \mathbb R$$ is simply the function mapping $$\Omega\ni\omega\mapsto g(X(\omega))$$. Thus if $$X_1$$ and $$X_2$$ are both random variables on the same probability space it is easy to talk about things like $$g\circ(X_1+X_2)$$. This is simply the map $$\omega\mapsto g(X_1(\omega)+X_2(\omega))$$.
For an example we can consider the most simple of experiments: flipping a fair coin twice. Then our sample space is $$\Omega=\{HH,HT,TH,TT\}$$, and our $$\sigma$$-algebra is simply the power set. The most natural probability measure is the one assigning a value of $$1/4$$ to each singleton in $$\Omega$$. Suppose we win a bet if in the experiment two heads occur. We can then capture this by the random variable $$X:\Omega\to \mathbb R$$ given by $$X(HH)=1$$ and $$X$$ is $$0$$ on all other elements of $$\Omega$$. Thus we see $$X$$ is simply a function, just not from $$\mathbb R$$ to itself.
Often in statistics we actually don't see $$\Omega$$ written out explicitly, because it often consists of weird things, like the population of the earth, but it is always exists underneath when we're doing probability calculations. In fact for a r.v. $$X$$ the CDF, $$F_X$$ is defined in terms of the probability space as follows $$F_X(x)=\mathbb P\{\omega\in \Omega:X(\omega)\leq x\}.$$


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# 2.  Let $\overline {PQ}$ be the perpendicular to the line segment $\overline {XY}$ . Let $\overline {PQ}$ and $\overline {XY}$ intersect in the point A. What is the measure of $\angle PAY$ ?

$PQ\perp XY$
$PA\perp AY$
Therefore $\angle PAY=90^{o}$

format: plain


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# nLab (infinity,2)-topos

Contents

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Higher topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

An (∞,2)-category of (∞,2)-sheaves. The joint higher generalization of the notion of (∞,1)-topos and 2-topos.

The archetypical example is the (infinity,2)-category of (infinity,1)-categories $Cat_{(\infty,1)}$. See also at formal category theory.

flavors of higher toposes

Last revised on October 13, 2021 at 12:01:58. See the history of this page for a list of all contributions to it.


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## Precalculus (6th Edition) Blitzer

We know that whenever a function $f\left( x \right)$ of degree $n$ is divided by another function $g\left( x \right)$ of degree $m$, the degree of the quotient must be $n-m$. Well, in the above case, the degree of $f\left( x \right)$ is $6$ when divided by $g\left( x \right)$ of degree $3$; the degree of the quotient must be $6-3=3$. But, the given degree of the quotient is $2$. Hence, the given statement is false.


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# Monthly Archives: May 2019

## Simple Sets and the Recursion Theorem

These notes on Simple Sets are a grabbag about the simple sets of recursion theory. If you don’t know what those are, you probably are not interested, but the Wikipedia article is nice and short and gives the basics.

The last result uses the “shiny black box” (see below), which seems like cheating, but isn’t!

I wrote up the notes sometime in the 1980s based on two papers, and on a lecture by Michael Stob at MIT (reporting on joint work with Maas and Shore). They discuss effectively simple sets and promptly simple sets.

Filed under Logic

## Why “atomic bomb” instead of “nuclear”?

Why do we (mostly) say atomic bomb instead of nuclear bomb, which is technically more correct? This was asked on the History of Science and Math stackexchange. Here’s my answer.

Filed under History, Physics

## Non-standard Models of Arithmetic 9

[In Post 21, Bruce Smith and I discuss the proof of the Paris-Harrington theorem.]

MW: Time to talk about the Paris-Harrington theorem. Originally I thought I’d give a “broad strokes” proof, but then I remembered what you once wrote: keep it fun, not a textbook. Anyway, Katz and Reimann do a nice job for someone who wants to dive into the details, without signing up for a full-bore grad course in model theory. So I’ll say a bit about the “cast of characters” (i.e., central ideas), and why I think they merit our attention.

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 8

JB: So, you were going to tell me a bit how questions about the universe of sets cast their shadows down on the world of Peano arithmetic.

MW: Yup. There are few ways to approach this. Mainly I want to get to the Paris-Harrington theorem, which Enayat name-checks.

First though I should do some table setting of my own. There’s a really succinct way to compare ZF with PA: PA = ZF − infinity!

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 7

MW: Our goal for the next few posts is to understand Enayat’s paper

• Ali Enayat, Standard models of arithmetic.

JB: Yee-hah!

MW: I’m going to take a leisurely approach, with “day trips” to nearby attractions (or Sehenswürdigkeiten, in the delightful German phrase), but still trying not to miss our return flight.

Also, I know you know a lot of this stuff. But unless we’re the only two reading this (in which case, why not just email?), I won’t worry about what you know. I’ll just pretend I’m explaining it to a younger version of myself—the one who often murmured, “Future MW, just what does this mean?”

Filed under Conversations, Peano Arithmetic

## Friedberg’s Enumeration without Duplication

Recursively enumerable (r.e.) sets are “semi-decidable”: if x belongs to an r.e. set W, then there’s a terminating computation proving that fact. But there may not be any way to verify that x does not belong to W. The founding theorem of recursion theory—the unsolvability of the Halting Problem—furnishes an r.e., non-recursive set. For this set, we have a program listing what’s in W, but no program listing what’s out.

Filed under Logic

## Non-standard Models of Arithmetic 6

JB: It’s interesting to see how you deploy various philosophies of mathematics: Platonism, intuitionism, formalism, etc. For a long time I’ve been disgusted by how people set up battles between these, like Punch-and-Judy shows where little puppets whack each other, instead of trying to clarify what any of these philosophies might actually mean.

For example, some like to whack Platonism for its claim that numbers “really exist”, without investigating what it might mean for an abstraction—a Platonic form—to “really exist”. If you define “really exist” in such a way that abstractions don’t do this, that’s fine—but it doesn’t mean you’ve defeated Platonism, it merely means you’re committed to a different way of thinking and talking.

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 5

MW: John, you wrote:

Roughly, my dream is to show that “the” standard model is a much more nebulous notion than many seem to believe.

and you gave a good elucidation in post 2 and post 4. But I’d like to defend my right to “true arithmetic” and “the standard model $\mathbb{N}$“.

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 4

MW: I wrote: “I don’t like calling the omega of a model of ZF a standard model, for philosophical reasons I won’t get into.”

JB: I like it, because I don’t like the idea of “the” standard model of arithmetic, so I’m happy to see that “the” turned into an “a”.

1 Comment

Filed under Conversations, Peano Arithmetic

## Non-standard Models of Arithmetic 3

[Reminder: JB=John Baez, MW=Michael Weiss.]

MW: Besides Kaye and Kossak & Schmerl., I should mention the book by Hájek and Pudlák, but I don’t have a copy of that. Thanks muchly for the Enayat paper, which looks fascinating.

What you and Enayat are calling the “standard” model of arithmetic is what I used to call “an omega”, i.e., the omega of a model of ZF. Is that the new standard terminology for it? I don’t like it, for philosophical reasons I won’t get into. (Reminds me of the whole “interpretations of QM” that books have to skirt around, when they just want to shut up and calculate.)

Leaving ZF out of it, a friend in grad school used to go around arguing that 7 is non-standard. Try and give a proof that 7 is standard using fewer than seven symbols. And of course for any element of a non-standard model, there is a “proof” of non-standard length that the element is standard. I think he did this just to be provocative. Amusingly, he parlayed this line of thought into some real results and ultimately a thesis.


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# Direction vector of a line

How would I find the direction vector of a line given in the form ax + by + cz +d =0?

Thankyou.

TD
Homework Helper
I see an equation of a plane, not a line.

Damn, sorry, got carried away with my letters. I meant ax + by +c = 0. Sorry about that...

TD
Homework Helper
Nevermore said:
Damn, sorry, got carried away with my letters. I meant ax + by +c = 0. Sorry about that...
For a line, (1,m) is always a directional vector where m is the slope. Can you find the slope?

Yeah - I thought it would be something simple like that. Thanks for your help.

TD
Homework Helper
You're welcome

Thanks for asking that Q, it helped me too

Defennder
Homework Helper
I don't think there's a need to thank someone who started a thread two years ago and have not posted since.

LORL.... Thanks for pointing that out. ( At least you still logon)


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How To Convert Euros To Pounds Manually

# How To Convert Euros To Pounds Manually

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## How To Convert Currency In Microsoft Excel

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dollars to British pounds (1 USD to GBP is in cell B2). When you change the absolute cell reference and use alternative data sources (like the FloatRates GBP data source to convert from GBP to other currencies), you can convert from any currency to ekhd.skechersconnect.com: Ben Stockton.

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## Whats The Formula For Converting Pounds To Euros With A

dollars to euros using a site like XE or Oanda. These sites are updated as the market changes. As of March\$1 equaled €   You convert one currency to another using the exchange rate between those two currencies. For example, an exchange rate of 1 US dollar to 2 euros means you convert 1 USD to 2 EUR and 2 EUR to 1 USD. It’s about how much one currency is worth in the other, use our currency converter to find out.

Convert 1 Euro to British Pound. Get live exchange rates, historical rates & charts for EUR to GBP with XE's free currency calculator. Instant free online tool for GBP to USD conversion or vice versa.

## How To Calculate The Exchange Rate [Step-by-Step Guide]

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## How To Calculate Euros Into Pounds

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text: So that’s why if any of the event probability is greater than 1, calculator shows an error message. For calculating single discrete random variables of Expected Value, one must multiply the value of the variable by the probability of that value occurring. 2 Biased/Unbiased Estimation In statistics, we evaluate the “goodness” of the estimation by checking if the estimation … The bias of the estimator is the difference between the true value of the estimator, and its expected value: $$\operatorname{Bias}(\bar{\theta}) = \operatorname{E}(\bar{\theta} - \theta)$$. Here's why. The population total = 16 ticks (i.e., 2+4+2+8= 16) for the 4 individual rats, then the population mean. Can you identify this restaurant at this address in 2011? expectation and variance. Variance A useful formula, where a and b are constants, is: E[aX + b] = aE[X] + b [This says that expectation is a linear operator]. Did my 2015 rim have wear indicators on the brake surface? Proof of variance of stationary time series, Calculate the variance of $\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n S(X_i - X_j)$ for $X_1,\ldots,X_n$ i.i.d. Provide this information, the calculation is very simple. If A is a vector of observations, the variance is a scalar.. Therefore, there is not a single possibility of having a probability greater than 1 in any event or total of all events. The variance of the estimator is. This expected value formula calculator finds the expected value of a set of numbers or a number that is based on the probability of that number or numbers occurring. MathJax reference. By calculating expected values, expected outcomes of probabilities are calculated by a set of numbers and the individual probabilities sum up to 1 or 100%. Step 2: Why is the unbiased sample variance estimator so ubiquitous in science? Having the OLS estimators in this form we can easily find the expected value and variance: The expected value of the OLS estimators. To find the combination of the values, we have Combination Calculator. By Developing 100+ online Calculators and Converters for Math Students, Engineers, Scientists and Financial Experts, calculatored.com is one of the best free calculators website. We try to estimate these values through a sample of 20 random oak trees. It produces a single value while the latter produces a range of values. Assume that $X_i$ has mean $\mu$ and variance $\sigma^2$ Random Variable gives its weighted average. Assume we have an estimator $\bar{\theta}$ for a parameter $\theta$. The bias of the estimator is the difference between the true value of the estimator, and its expected value: ( θ ¯) = 0, then θ ¯ is called unbiased. This lecture presents some examples of point estimation problems, focusing on variance estimation, that is, on using a sample to produce a point estimate of the variance of an unknown distribution. The expected value of the estimator is equal to the true mean and is therefore unbiased: The proof is the same found in the previous example. Algorithm for simplifying a set of linear inequalities. For example, five players playing spin the bottle. Point estimation is the opposite of interval estimation. Practical example. For finding only the leftover value, use the Remainder Calculator or if the value exceeds to its limits/values, try out Rounding Calculator. Provide this information, the calculation is very simple X=Number of Players=1,2,3,4,5 Weighted Average=(15∗1)+(15∗2)+(15∗3)+(15∗4)+(15∗5) Weighted Aver… By squaring every element, we get (1,4,9,16,25) with mean 11=3²+2. \operatorname{var}\left( \frac 1 N \sum_{i=1}^N X_i\right) = \frac 1 {N^2} \operatorname{var}\left( \sum_{i=1}^N X_i \right) = \frac 1 {N^2} \sum_{i=1}^N \operatorname{var}(X_i) = \cdots Give the number of the probability of success and values of x, expected value calculator will notify you about the expected value for a discrete random variable. But which identity i need to use for calculating the variance?? Consiste The expected value of the variable 1 − X is the mirror-symmetry of the expected value based on X: E  [ 1 − X ] = β α + β E  [ X ( 1 − X ) ] = E  [ ( 1 − X ) X ] = α β ( α + β ) ( α + β + 1 ) {\displaystyle {\begin{aligned}&\operatorname {E} [1-X]={\frac {\beta }{\alpha +\beta }}\\&\operatorname {E} [X(1-X)]=\operatorname {E} [(1-X)X]={\frac {\alpha \beta }{(\alpha +\beta )(\alpha +\beta +1)}}\end{aligned}}} the sample mean), \begin{align} So beta has unbiased, it's expected value is what its like to estimate, what we'd like to estimate. Why weren't Tzaddok and Baytos put to death? 00:17 Wednesday 16th September, 2015 Estimates of the variance, and standard deviation, of autocorrelated data will be biased. What's the difference between 「お昼前」 and 「午前」? Bias can also be measured with respect to the median, rather than the mean, in which case one distinguishes median-unbiased from the usual mean-unbiasedness property. By calculating expected value, users can easily choose the scenarios to get their desired results. Example 2: Let X1;X2;¢¢¢;Xn be i.i.d. All rights reserved. How many computers has James Kirk defeated? Click the "Calculate" button and the results will represent the expected value. The expected value of a constant is just the constant, so for example E(1) = 1. Is it illegal to market a product as if it would protect against something, while never making explicit claims? B. the sample variance can be estimated. Bias is a distinct concept from consistency. To learn more, see our tips on writing great answers. We hope you liked this article and the functionality of our expected value calculator. For instance, set (1,2,3,4,5) has mean 3 and variance 2. How can I show that a character does something without thinking? Expected value and variance-covariance of generalized hyperbolic distributions. Loading ... ESTIMATION OF MEASUREMENT UNCERTAINTY IN … = {} & \operatorname{E} \left( \frac 1 N \sum_{i=1}^N X_i \right) - \mu \\[10pt] We compute the expectation and variance of the sample variance when the random variables are generated from a normal distribution. Enter all values numerically and separate them by commas. intuitively, the mean estimator x= 1 N P N i=1 x i and the variance estimator s 2 = 1 N P (x i x)2 follow. Distribution of the estimator & \operatorname{E}(\bar{\mu})-\mu \\[10pt] The function mean returns the expected value. Are there any drawbacks in crafting a Spellwrought instead of a Spell Scroll? An estimator or decision rule with zero bias is called unbiased. Please provide your valuable comments. Can you derive the distribution of the Gaussian variance estimator without using moment generating functions? For example, five players playing spin the bottle. Multiplying a random variable by a constant multiplies the expected value by that constant, so E[2X] = 2E[X]. I have to prove that the sample variance is an unbiased estimator. Step 1: bias, variance and mean squared error of the following mean estimators: (a) $\bar{\mu} =\frac 1 N \sum_{i=1}^N X_i$(i.e. = 16/4 = 4. For weighted average calculations, try Average Calculator. Thanks for contributing an answer to Cross Validated! The variance of the estimator $\bar{\theta}$ is the expected square deviation from its expected value: $$\operatorname{var}(\bar{\theta}) = \operatorname{E}[(\bar{\theta} − E[\bar{\theta} ])^2]$$, The mean squared error of the estimator $\bar{\theta}$ is, $$\operatorname{Error}(\bar{\theta}) = \operatorname{E}(\bar{\theta} − \theta)^2 = \operatorname{Bias}(\bar{\theta})^2+\operatorname{var}(\bar{\theta})$$. Expected value and bias Recall that E[ ijX i] = 0, so 1 n Xn i=1 (x i x )E[ i] = 0 (23) Thus, E h ^ 1 i = 1 (24) Since the bias of an estimator is the di erence between its expected value and the truth, ^ 1 is an unbiased estimator of the optimal slope. Also in this case the proof is the same found in the previous example. Xi = All Possible Outcomes. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. See Chapter 2.3.4 of Bishop(2006). Do Magic Tattoos exist in past editions of D&D? Variance of the estimator. This online expected value calculator will help you to find the expected value swiftly and easily of a discrete random variable X. By using this calculator, you will get detailed solutions to your problems. Variance and standard deviation As with the calculations for the expected value, if we had chosen any large number of weeks in our estimate, the estimates would have been the same. Measures center of the Probability distribution, Reduces information to one possibility /answer, The risk rate is high as it ignores the risk, The difficulty for assessing probabilities of different results. P(Xi) = Probability Every time the total possible result is 100%. Hence, the mean value of the sample estimators equals the population parameters. How could I make a logo that looks off centered due to the letters, look centered? Variance Simple English Wikipedia the free encyclopedia. To calculate expected value, with expected value formula calculator, one must multiply the value of the variable by the probability of that value is occurring. Enter all known values of X and P(X) into the form below and click the "Calculate" button to calculate the expected value of X. Click on the "Reset" to clear the results and enter new values. Below is my code to compute the expected value and variance. These are the statistical parameters of the entire population. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. If we cannot complete all tasks in a sprint. In Brexit, what does "not compromise sovereignty" mean? If A is a multidimensional array, then var(A) treats the values along the first array dimension whose size does not equal 1 as vectors. I've created my own function using the definitions of expected value and variance and then compare against matlab's mean and var functions. Most importantly this value is the variables long-term average value. Do they emit light of the same energy? The unbiased estimator for the variance of the distribution of a random variable , given a random sample is That rather than appears in the denominator is counterintuitive and confuses many new students. To accurately find the mean value of a set of values, we introduce the Mean Values Calculator . It is easy to check that these estimators are derived from MLE setting. The formula used to find the expected value for a number or set of numbers is defined as : Expected value = Thus, the expected value of the estimator^^.. is 4; this is denoted as E(). You should confirm these steps your self. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. This unbelievable library created by Sebastian Raschka provides a bias_variance_decomp() function that can estimate the bias and variance for a model over several samples. Sum of its associated probability * All possible outcomes, EV = Expected Value of an Opportunity ...gave me (the) strength and inspiration to. Point estimation of the variance. If A is a matrix whose columns are random variables and whose rows are observations, V is a row vector containing the variances corresponding to each column.. The variance can be calculated using a variance estimator, e.g. ... Is the expected value we sample in TD-learning action-value Q or state-value V? For only finding the center value, the Midpoint Calculator is the best option to try. It is easy to learn to find the expected value. Why is "issued" the answer to "Fire corners if one-a-side matches haven't begun"? • The unbiased sample variance estimator is 13 ... • Bias measures the expected deviation from the true value of the function or parameter • Variance provides a measure of the expected deviation that any particular sampling of the data is likely to cause 18 . random variables. On the other hand, interval estimation uses sample data to calcul… Triangular Distribution: Expected Value, Variance, and Quantile Function Ashkan Negahban. = {} & \frac 1 N \sum_{i=1}^N \operatorname{E}(X_i) - \mu \\[10pt] Copyrights 2020 © calculatored.com . by Marco Taboga, PhD. You can also use our other calculators. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. \end{align}. While the expected value of x_i is μ, the expected value of x_i² is more than μ². In statistics, "bias" is an objective property of an estimator. Expected Value is calculated for single discrete variables, multiple discrete variables, single continuous variables, and multiple continuous variables. It only takes a minute to sign up. Expected value formula calculator does not deals with significant figures. And then also we can calculate the variance of beta hat under these assumptions. A point estimator is a statistic used to estimate the value of an unknown parameter of a population. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. How do you know how much to withold on your W2? . Recall that it seemed like we should divide by n, but instead we divide by n-1. Calculate the rev 2020.12.8.38142, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…, Asymptotic distribution of sample variance of non-normal sample, Comparison of two estimators based on mean squared error. ... gave me ( the expected value of variance estimator strength and inspiration to single continuous variables, and multiple variables. Seemed like we should divide by n-1 the Gaussian variance estimator so ubiquitous in science can expected value of variance estimator show a! ( Philippians 3:9 ) GREEK - Repeated Accusative Article like to estimate these values through a of. X in blank shaded boxes logo that looks off centered due to letters! Prejudice '', what does Darcy mean by  Whatever bears affinity cunning... For you a is a statistic used to calculate significant figures same found in the multivariate.. Using expected value of variance estimator any point in the univariate case and the results will represent the expected swiftly. Pride and Prejudice '', what we 'd like to estimate, what does Darcy mean . Function Ashkan Negahban constant, so for example, five players playing spin the bottle wear indicators on brake... Off centered due to the letters, look centered or personal experience my 2015 rim have wear indicators the. Where the increment of larger numbers is greater than 100 % data when calculating a single possibility having! Sample data when calculating a single statistic that will be biased 1,4,9,16,25 with... An infinite number of times, you will get detailed solutions to your problems a value... 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text:
# A problem of counters, timing and etoolbox

I am attempting a format for recipes. For input, each step of the method is accompanied by the ingredients for that step. The output places all of the ingredients together, and all of the steps of the method together, in what is commonly termed the “standard” format. Numbers are attached to the method steps that match the numbers for the ingredients, enabling a convenient correspondence between the two. I have tried various incantations involving \xdef and \xappto but seem to be missing an important piece of the puzzle. There is this but it uses collect.sty and I don't know how to apply that here. Clarification: I don't mean to imply that I want to use collect.sty here, but that I don't know that solution applies to this problem.

\documentclass[]{article}

\usepackage[papersize={5.5in,8.5in},margin=0.6in,bottom=0.7in]{geometry}
\usepackage{fontspec,xparse,etoolbox}

\newcounter{stepnum}

%% |=====8><-----| %%

\begingroup
\lccode~=\^^M
\lowercase{%
\endgroup
\NewDocumentCommand{\ingrline}{u{ }u{ }u{~}}{% #1=measure; #2=unit; #3=ingr. name
\xappto{\ingrbody}{\noexpand\ing{#1}{#2}{#3}}~
}%
\def\methline#1~{% Line of method text
\xappto{\methbody}{\unexpanded{\mth{#1}}}~%
}%
}

%% |=====8><-----| %%

\NewDocumentCommand{\ing}{mmm}{% Format: #1=measure; #2=unit; #3=ingr. name
\textbf{\xstepnum}:\ \textit{#1-#2}\ \emph{\textbf{#3}}\par%
}

\NewDocumentCommand{\mth}{m}{% Format: Line of method text
\textbf{\xstepnum:\ }#1\par
}

\NewDocumentEnvironment{recipe}{s O{} m}{% *=?; #2=keyval; #3=title
\setcounter{stepnum}{0}%% stepnum attaches both to group of ingredients and each method step
\gdef\ingrbody{}%
\gdef\methbody{}%
}{%
%% output \ingrbody and \methbody
\ingrbody
\par
\methbody
}

\NewDocumentEnvironment{step}{}{%
\stepcounter{stepnum}
\xdef\xstepnum{\thestepnum}
%% Assemble ingredients
\begingroup %% write ingredients to \ingrbody
\offinterlineskip
\obeylines
\everypar={\ingrline}
}{%
\endgroup %% End method
}

\NewDocumentCommand{\method}{}{%
\endgroup %% end ingredients
\begingroup %% begin method write method to \methbody
\offinterlineskip
\obeylines
\everypar={\methline}
}

%% |=====8><-----| %%

\setmainfont{STIX Two Text}

\pagestyle{empty}

\begin{document}

\begin{recipe}{A test}
\begin{step}% 1
1 cup butter (should be 1:'')
½ cup sugar (should be 1:'')
\method
Beat butter and sugar together until fluffy. (should be 1:)
\end{step}
\begin{step}% 2
2 cups flour
1 tablespoon baking powder
\method
Add to the butter mixture and combine.
\end{step}
\end{recipe}

\end{document}

gives this output:

The final format will be very different. This is just an MWE.

• Counters are always global (count registers are local). Nov 20, 2021 at 16:01
• Somewhat related: tex.stackexchange.com/questions/301298/… Nov 20, 2021 at 16:05
• @JohnKormylo I think I was--at some level--aware of that, but I don't see how that applies to this problem... Nov 20, 2021 at 19:57

You globally assign to \xstepnum the value of the stepnum counter right after the environment step is called and after the counter has been increased. But the lines to be finally printed are put together (and the relvant macros expanded) only when the recipe environment is closed. At this time, however, \xstepnum holds the number of the last step. Therefore this number will be put in every entry or line.

Therefore, you need to pass the current step through the whole macro chain instead, up to the point where you put together the final lines, like this:

\documentclass[]{article}

\usepackage[papersize={5.5in,8.5in},margin=0.6in,bottom=0.7in]{geometry}
\usepackage{fontspec,xparse,etoolbox}

\newcounter{stepnum}

%% |=====8><-----| %%

\begingroup
\lccode~=\^^M
\lowercase{%
\endgroup
\NewDocumentCommand{\ingrline}{mu{ }u{ }u{~}}{% #1=stepnum; #2=measure; #3=unit; #4=ingr. name
\xappto{\ingrbody}{\ing{#1}{#2}{#3}{#4}}~
}%
\NewDocumentCommand{\methline}{mu{~}}{% #1=stepnum; #2 rest
\xappto{\methbody}{\mth{#1}{#2}}~%
}%
}

%% |=====8><-----| %%

\NewDocumentCommand{\ing}{mmmm}{% Format: #1=stepnum; #2=measure; #3=unit; #4=ingr. name
\textbf{#1}:\ \textit{#2~#3}\ \emph{\textbf{#4}}\par%
}

\NewDocumentCommand{\mth}{mm}{% Format: #1=stepnum, #2=Line of method text
\textbf{#1:\ }#2\par
}

\NewDocumentEnvironment{recipe}{s O{} m}{% *=?; #2=keyval; #3=title
\setcounter{stepnum}{0} %% stepnum attaches both to group of ingredients and each method step
\gdef\ingrbody{}%
\gdef\methbody{}%
}{%
%% output \ingrbody and \methbody
\ingrbody
\par
\methbody
}

\NewDocumentEnvironment{step}{}{%
\stepcounter{stepnum}
%% Assemble ingredients
\begingroup %% write ingredients to \ingrbody
\offinterlineskip
\obeylines
\everypar={\ingrline{\thestepnum}}
}{%
\endgroup %% End method
}

\NewDocumentCommand{\method}{}{%
\endgroup %% end ingredients
\begingroup %% begin method write method to \methbody
\offinterlineskip
\obeylines
\everypar={\methline{\thestepnum}}
}

%% |=====8><-----| %%

\setmainfont{STIX Two Text}

\pagestyle{empty}

\begin{document}

\begin{recipe}{A test}
\begin{step}% 1
1 cup butter (should be 1:'')
½ cup sugar (should be 1:'')
\method
Beat butter and sugar together until fluffy. (should be 1:)
\end{step}
\begin{step}% 2
2 cups flour
1 tablespoon baking powder
\method
Add to the butter mixture and combine.
\end{step}
\end{recipe}

\end{document}

PS: In the definition of \ing I changed \textit{#1-#2} to \textit{#1~#2} since I believe that this is what you actually meant.

• Ah. I could see what was happening, not how to fix it. Fiat lux! Nov 20, 2021 at 20:58