Questions tagged [function]

A mapping between a set of inputs and a set of outputs.

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Is $f(\widehat{x})$ ($\widehat{x}$ is a random sample of $x$) equivalent to $\widehat{f(x)}$ (a random sample of $f(x)$)? [closed]

Assuming $f(x)$ is an injective function, is sampling $\widehat{x}$ from the distribution of $x$ then calculating $f(\widehat{x})$ equivalent to directly sampling from the distribution of $f(x)$? Note ...
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Find the relationship among adjusted 3-pointer (not parameter, but formula like Pythagenport, Pythagenpat), goals scored and goals allowed in football

I was seeking a formula to relate goals scored and goals allowed to points. Firstly, I worked with points percentage — points divided by available points, which I set to the number of matches ...
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Is sequence of probability mass functions always uniformly bounded

Say that we have a sequence of discrete random variables, $\left\{X_n\right\}_{n \in \mathbb{N}}$, which converges to a random variable, $X$, with a continuous distribution, e.g., the Normal (Gaussian)...
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Should 3-pointers be worth 2.5 points?

I was seeking an alternative scoring rule sets instead of three points for a win (gained more engaging and balanced) to cancel the theory "Banking a draw meaning a new kind of loss". It's ...
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Finding the derivative of the kSVM classifier funcion with respect to the weight vector?

I would like to try a different approach for defining the kSVM. However to do that at some point I need a derivative of the clasifier function $y(\textbf{x})$ with respect to the weights vector ...
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Exploring Alternative Pythagorean Expectation Formulae

In a regular match, a team score $a_{i}$ goals and allow $b_{j}$ goals, both are natural numbers). The result will depend on: $$\left | a- b \right |\left\{\begin{matrix} \geq e. 2.0\,{\rm points}\,{\...
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Using output of categorical distribution in Pymc3 as index for selection of down stream parameters

The code that I would like to generate is something like: I want to have a categorical variable that maps to another value different from the index given by the categorical distribution, and fits as a ...
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How do you get this bifurcation diagram of this 2D discrete time model?

I want to replicate the results of "Chaos in Duopoly Pricing" by T. Puu, which can be found here: https://www.sciencedirect.com/science/article/pii/096007799190045B#section-cited-by. The ...
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How to find expected value from cumulative distribution function?

Hello everyone, I'm currently doing research based on the model in Competitive fit-revelation sampling and mixed pricing strategy (Wu &Deng, 2021). And I don't understand how they can conclude the ...
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Analytic structure of the Dirichlet L functions with the Hurwitz zeta function

I am getting confused about how to get results from below by using the Hurwitz zeta function. Really don't know how to do it. "The Dirichlet L functions for non-principal Dirichlet characters are ...
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How do you make a generalized function for the probability of choosing a specific amount of special objects by swapping objects between groups?

I'll start by describing a very specific scenario. Suppose there are 15 distinct objects, 4 of which are special in some unknown way. I have to select 4 of the 15 objects simultaneously, not knowing ...
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Determine expected goals from two normal distributions

Assume Team A scores an average of 3 goals per game with a standard deviation of 1.0, and assuming Team B allows an average of 2 goals per game with a standard deviation of 0.5. How would you ...
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Problem formulation classification task

I would like to know if it is correct for a classification task in a supervised learning to say the model we are looking for is a function from RxR to a discrete space $$ f:\mathbb{R}\times\mathbb{R} \...
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SMOTE vs Stratified Sampling in highly imbalanced dataset - classification

I am working on a project with the goal of predicting Cerebral strokes from brain arteries data (speed of blood, resistance etc. of one artery and of the neighboring ones). I have a dataset with ...
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Kaplan Meier estimator which form to use

I'm seeing different forms of kaplan meier estimator. The most intuitive one for me is $$ \hat{S}(t) = \prod_{i=1}^{j}(1-\hat{h}_{i}) $$ but my textbook uses $$ \hat{S}(t) = \prod_{r=1}^{i}\frac{n-r}{...
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Difference between Predictive Inference and Causal Inference

I am looking for functional mathematical notation to explain the difference between Predictive Inference and Causal Inference? I list an example model. I also list links further down that give ...
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Equal intervals on log scale

I want to make a plot in which the horizontal axis is log scaled, ranging from 1 to 1000. I want to divide the horizontal axis into $n$ equal intervals, where "equal" means that it appears ...
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distance between regression models

Consider two multivariate linear regression models (vector inputs and outputs) with the same domain observations. Namely, let: $X \in \mathcal{R}^{a \times N}$ be a matrix of domain observations (...
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Mapping a value spanning several years to one specific year

I'm trying to think how to best deal with this problem. I need to report infant mortality rates for a country per decade. So like this: Infant mortality rate in country X, per decade Year Value 1970 ...
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How to calculate exponential growth when the value is a percentage or probability?

I have a variable x that takes values between 1 and 0 (0<x<1). I want to calculate value of ...
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Property of unbiased estimators

If $f(x)$ and $f(y)$ are both unbiased estimators of $\mu$, aka $E[f(x)]$ = $E[f(y)]$ = $\mu$, is it possible that $f((x+y)/2)$ is also an unbiased estimator of $\mu$? We know $f((x+y)/2)$ would be ...
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statistical model to anlyse each entry with equal weight? and help in r function to do this?

We have conducted an experiment using partial replicated design in 4 replications(blocks). Each block has 84 entries (genotypes) where 12 entries are duplicated (to control environmental variation ...
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Function fit to skewed data and non-zero beginning of the function

I would like to find a function that would represent the best fit to represent this type of biological data. More precisely, I would like to estimate expected daily egg production by an insect, based ...
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Prove that every linear regression predictor is a linear function

A function $f : \mathbb{R}^D \rightarrow \mathbb{R}$ is linear if both of the following conditions hold. (1) For all $\textbf{x}, \textbf{y} \in\mathbb{R}^D, f(\textbf{x} + \textbf{y}) = f(\textbf{x}) ...
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How to use statistics to speed up row-wise computations on a data.frame?

I have a data frame with 10,000 rows and 40 columns. I am trying to apply a function to each of these rows. For each row, I am expecting to return a scalar which is the value of the statistic I am ...
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Polynomials that converges pointwise to a simple function on (-1,1) and bounded by $e^{|x|}$?

I am trying to prove a theorem related to the moment generating function. I will need a sequence of polynomial that converges to a simple function $K_{(-1,1)}(x)$ pointwise on the real line while ...
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How to weight ranks?

I have a set of responses ranked by my participants. For example, they gave responses A, B, C and ranked them as 3, 2, 1 (or C, B, A). I computed relative frequencies of each responses (A, B, C) and ...
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What is the equation of a line fitting a log-log model computed in R?

I am currently stuck, wanting to extract a line function from my fitted line on my log-log model. ...
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Why does non-parametric approach break down when the joint distribution is estimated by a finite data sample?

I am currently reading the paper on Gradient Boosting Machines - J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” Ann. Stat., vol. 29, no. 5, pp. 1189–1232, 2001, doi: 10....
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Why the output of softmax can be saturated when input differences become extreme? [duplicate]

Like sigmoid function that can be saturated when inputs are extremely different. But how these inputs can cause this function to be saturated?
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replacement error when feeding data from function into matrix in R [closed]

I am trying to write a function that creates simulated datasets. The function takes the argument size, being the size of a group, and it is meant to produce a matrix with 100 columns with values 0 or ...
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Other functions with sigmoid shape [duplicate]

Apart from the classic logistic or sigmoid function, are there other interesting functions that map real x into a positive space? Maybe they also have an S shape. Maybe inspired by some nature ...
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3 votes
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$f$ is a decreasing function whose integral converges. Does $\lim_{x \to \infty}xf(x) = 0$?

My finals are over and I cannot help but ruminate over this particular problem. Could anyone help prove this? Suppose $f$ is a continuous decreasing function on $[0,\infty)$ and $\int_0^\infty f(t)\, ...
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1 answer
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How to find parameters of sigmoid function given two thresholds?

Let's say I start with a sigmoid function $$f(x) = \frac{1}{1+\exp{\frac{-(x-c)}{d}}}$$ where the upper limit is 1 and lower limit 0, how can I for example find the appropriate values of the ...
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Which type of sums of squares does lm-function in R use?

I ran a two-way ANCOVA in R: ancova = lm(DV ~ IV1*IV2 + CV1 + CV2 + CV3, data = Data) summary.aov(ancova) Anybody know if this uses type III sums of squares? I ...
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1 vote
1 answer
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What is the 'feature function'? [duplicate]

The term 'feature function' is very frequently used in the context of machine learning, but I'm still not sure what it really is. Could anyone give the precise definition? Can it be understood as a ...
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Get Continuous Distribution from Discrete Variable: Problem 6.77 of Wackerly, Mendenhall, Schaeffer, 5th Ed

Problem Statement: $\newcommand{\szdp}[1]{\!\left(#1\right)}$ Let $v$ denote the volume of a three-dimensional figure. Let $Y$ denote the number of particles observed in volume $v,$ and assume that $Y$...
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Calculate probability density by function application

let's assume we have a probability density function which is $$f(X) = \frac{a}{x^a+1}$$ how can we calculate something like this: $$Y = ln(X)$$
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ML approach to learn function as output

I have a question on how to use ML (machine learning) methods for the following task. I have a m-dimensional (field is always over real numbers $R$) input vector $\vec{x}$, and I want to learn the ...
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2 votes
1 answer
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Is there a name for $\sum P(x) \frac{P(x)}{Q(x)}$ ? (P and Q are pmf)

I know that $\sum P(x) log \left( \frac{P(x)}{Q(x)} \right)$ is the kl-divergence. I'd like to know if there is a name for $\sum P(x) \left( \frac{P(x)}{Q(x)} \right)$ (no log), but couldn't find one. ...
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Relationship between deterministc function of random variables

Given a discrete $P(X,Y,Z)$ let's call $\Omega$ the set of all deterministic functions $f: XYZ \rightarrow W$ and $\Omega'$ the set of all deterministic functions $f': XY \rightarrow V$. Is it correct ...
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Problem with user-written function: "Error in shapiro.test(a) : is.numeric(x) is not TRUE" [closed]

Dear StackExchange community, I am trying to write a function to automate some normality-checks. Here is my try (automates Shapiro-Wilk test, histogram and Q-Q plot creation ): ...
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2 votes
2 answers
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How to make the best final prediction of the optimum value in a Bayesian optimization process?

I'm trying to understand the process of Bayesian optimization of a black box function and the bit I'm confused about is how to make the very last prediction of the true maximum after you have made all ...
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Fitting model to Wood's lactation curve

I'm trying to determine how I can estimate the following model: $y = at^bexp(ct)$, where $a$, $b$ and $c$ are estimate from a set of data. But I can't figure out how to derive the formula into ...
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How is an 'ogival function' defined?

Reading on a paper on factor analysis and measurement invariance I find the description of some functions as 'ogival' functions. In Google I find it referenced mostly in papers from the '70s and '80s....
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-1 votes
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Calclulating and creating x matrix using set.seed function [closed]

I have been asked to evaluate the sum of xi's? what could be wrong with the following code? How to correct it to work? set . seet (1) #set random seed x = rnorm (50) for ( j in 1:50){ s = s + x [ i ] ...
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Viterbi Algorithm [closed]

Can someone explain why Is it because for the left argument, we would find the most likely sequence of states given observations and for the right argument, we would eventually find the most likely ...
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A space of functions and their Fourier Transforms?

Conjugate variables and the Fourier transform are often used to analyze different states of a single object. For example in Quantum Mechanics it can be used to describe changing information about ...
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A function of random variables $X_1, ..., X_k$ that goes from $\mathcal{R}^k$ to the reals is measurable with respect to $\sigma(X_1, ..., X_k)$

I'm reading Resnick's "A probability Path" and doing exercise 3 on page 85. The statement is: Suppose $f : \mathcal{R}^k \rightarrow \mathcal{R}$ and $f \in \mathcal{B}(\mathcal{R}^k) / \...
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Do two dependent variables have a functional relationship?

Suppose $X$ and $Y$ are two dependent random variables (e.g. they are the elements of a bivariate normal distribution with $\rho\ne0$). Is it true that there always exists a function $f$ such that $f(...
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