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Questions tagged [approximation]

Approximations to distributions, functions, or other mathematical objects. To approximate something means to find some representation of it which is simpler in some respect, but not exact.

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15 views

Poisson distribution question

An airline has found that the number of people booked on flights who do not arrive at the airport follows a Poisson distribution at the rate of 2% per flight.For a flight with 146 seats ,150 are sold ...
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How can I apply the Poisson ($\mu$) distribution to two series of random draws?

I have the following question: A box contains 1000 balls, of which 2 are black and the rest are white. If two series of 1000 draws are made at random from this box, what approximately, is the chance ...
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Penalize the MSE of half of predicted values more than the other half

I'm using MSE loss for an multi-layer perceptron that learns to approximate the target feature vector $[\hat{x_1}, \dots, \hat{x_N}, \hat{y_1}, \dots, \hat{y_N}]^\intercal$. The catch here is that I'd ...
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How to get better approximation than Central Limit Theorem

This is continuation of my problem Calculate variance of sum random variables Suppose random variable $X$ takes 3 values $1, 2, 3$ with probability $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$. ...
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How to approximate the distribution of the sum of multiple multinomial random variables?

Say we have $T$ independent Multinomial random variables $X_1,X_2\dots X_T$, with $p(X_t=m)=p_{t,m},m\in\{1,2,...M\}$. What would be the distribution of $X_1+X_2+...+X_T$? If there is no closed-form, ...
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Approximation for the sampling error of the number of positive cases in a Bernoulli trial

Reading the book "Energy for Future Presidents" I found a way of approximating the binomial proportion sampling error which I never saw before, and I would like to know if my derivation is correct. ...
69 views

Inverse-normal CDF approximation in Excel, Python or R

I read that the implementations of Inverse-normal cumulative distribution function (CDF) /quantile / ppf in R, Python (scipy) and Excel give similar results. However, I can't find the very formulae ...
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Order of continuity of an ANN approximation dependent on the activation functions used?

If I have understood this correctly, a result from Hornik et al.'s Universal Approximation of an Unknown Mapping and Its Derivatives Using Multilayer Feedforward Networks essentially states that, if ...
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Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
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at what probability will the probability we start considering the data?

For example I have this problem, Do Americans tend to vote for the taller of the two candidates in a presidential election? In 30 presidential elections since 1856, 18 of the winners were taller than ...
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Approximation of copulas

I'm studying copulas, finished the Introduction to Copulas by Nelsen. I'm interested in the latest/best known/etc approaches for approximating any Copula, or some families of copulas, so would be ...
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Approximating the first moment of $h(x)$ where $x$ ~${\rm log\,normal}(\mu, \sigma)$

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$)? So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
38 views

Compute Mean of Normal Distribution where x% of Values are over y

I am looking for a way to determine the mean of a normal distribution (with given variance), where e.g. $z = 0,37 = 37\%$ of values should be above a certain value $a$ (e.g. 0,2)? My first idea was ...
301 views

How state aggregation in reinforcement learning works?

I am watching Prediction with linear approximation video course in the RL class by prof. Sutton. He presented state aggregation approach on a random walk problem. It seems that this approach just ...
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Approximation or closed form equation for summation of logistic function [closed]

The spread of epidemics follows a logistic growth, given in the equation below $I(t) = \frac{N}{1+(N-1)exp^{-rNt}}$ where, N is the population size, r is infection rate, t is time , I(t) is ...
256 views

The Universal Approximation Theorem vs. The No Free Lunch Theorem: What's the caveat?

The universal approximation theorem: A neural network with 3 layers and suitably chosen activation functions can any approximate continuous function on compact subsets of $R^n$. The no free ...
109 views

Approximating the error of maximum likelihood estimation

I have a log likelihood function of a model and I want to find $\mu$ and $\sigma^2$ which maximize the log likelihood. Since the log lik function is quite complex, I decided to use Nelder-Mead ...
153 views

In exactly what sense do MCMC draws approximate the target?

Background We want to sample from some intractable density $\pi(\theta)$. Using an MCMC algorithm, we generate a sample of draws $\{\theta_i\}_{i=1}^N$ from a Markov chain that has $\pi(\theta)$ as ...
60 views

Optimization: Approximate function - Which points to evaluate next?

I am looking for a statistical method (and a link to a nice R package would be cool too!) which allows me to find which point to evaluate next for a given function. I have a non-stochastic function z ...
372 views

Normal approximation to Poisson: With Continuity Correction the Approximation Seems Worse

This is Exercise 3 in Section 6.3 of Probability and Statistics, 4th edition, by DeGroot and Schervish: Suppose that the distribution of the number of defects on any given bolt of cloth is the ...
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Analytically solving sampling with or without replacement after Poisson/Negative binomial

Short version I am trying to analytically solve/approximate the composite likelihood that results from independent Poisson draws and further sampling with or without replacement (I don't really care ...
I have over $18000$ curves that I need to compress to save $\geq 50\%$ of space. Each curve is described by points $f(1), f(2), ..., f(96)$, each $f(x)$ is 8-bit long. The curves in compressed form ...
Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...