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Questions tagged [calculus]

For statistical questions involving calculus. Please use also a more statistical tag. For purely mathemathical questions about the calculus, it is better to ask at math SE https://math.stackexchange.com/

Wikipedia has an article https://en.wikipedia.org/wiki/Calculus with further references.

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Under the 0-1 loss function, the Bayesian estimator is the mode of the posterior distribution

My notes are rather light when it comes to this topic. I understand that the bayesian estimator, defined as (for sample space $\hat{x}$): $E[\Theta | \hat{x}] = \int_{ \forall \Theta}yf_{\theta|\hat{x}}(y|\hat{x})dy $ (ie. the mean of the posterior…
mrhappysmile
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Deriving linear regression gradient with MSE

So I've been tinkering around with the backpropagation algorithm and to try to get a better understanding of how it works and my calculus is quite rusty. I've derived the gradient for linear regression using a MSE loss function, but have nowhere to…
Pavlin
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gradient versus partial derivatives

how exactly is partial derivative different from gradient of a function? In both the case, we are computing the rate of change of a function with respect to some independent variable. While I was going through Gradient Descent, there also the…
Upendra01
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Limit of Integration of continuous function

How to evaluate the following limit- $$\lim_{n \to \infty} \int_0^1 \int_0^1\cdots\int_0^1 f \bigg(\frac{x_1 + x_2 + \cdots + x_n}{n} \bigg) dx_1 dx_2....dx_n$$. Here $f()$ is a continuous function $f:[0,1] \to \mathbb{R}$. Is there any bounds for…
edison
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Why is optimisation solved with gradient descent rather than with an analytical solution?

I'm trying to understand why, when trying to minimise an objective function, gradient descent is often used, rather than setting the gradient of the error to zero, and solving it analytically. In school, I was taught that in order to minimise a…
Karnivaurus
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Correlation between normal and log-normal variables

(This is not a homework question.) Let $(X_1 \sim N(\mu_1,\sigma_1), X_2 \sim N(\mu_2, \sigma_2))$ be a bivariate normal random variable with the correlation between $X_1$ and $X_2$ given by $\rho$. Let $Y_i = \exp(X_i)$ be a log-normal variable.…
Will Bradshaw
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Derivation of M step for Gaussian mixture model

Summary So to summarize my question, how can I take \begin{align} = \sum_{i=1}^{n}W_{i1} \left(log (1-\sum_{j=2}^{K}\pi_j) -\frac{1}{2} log(|\Sigma_1|) -\frac{d}{2} log(2\pi) -\frac{1}{2}(x_i-\mu_1)^{T} \Sigma_{1}^{-1}(x_i-\mu_1) \right)+ …
Stochastic
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Why does Judea Pearl call his causal graphs Markovian?

In his texts on causality, Judea Pearl always refers to the simplest graphs he uses, i.e. the acyclic graphs with independent confounders, as Markovian. I don't see why these graphs contain anything like a Markov-property.
Abelaer
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Finding the slope at different points in a sigmoid curve

This is my data. x <- c(0.5,3.0,22.2,46.0,77.3,97.0,98.9,100.0) plot(x, pch = 19) I want to fit a curve through these points and then calculate the slope at different points. Could anyone tell me how to do this in R EDIT Previously I had fitted a…
89_Simple
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How do you obtain the standard error for a slope at a given data point, for curvilinear regression?

A distribution looks like this: modeled by an equation $y=1.0333x^2 - .5382x + 1.6905.$ Find the rate of change (i.e. the slope at that point of the regression equation) at point 6 (the x axis value), and give the standard error of that slope.…
Tom
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When will $\mathbb{E}[g(S_n/n)]$ exist given $\mathbb{E}[g(X_1)]$ exists?

Suppose $X_1, X_2,..., X_n$ are i.i.d. random variables with distribution $\pi$ on some probability space. Let $g$ be a measurable function such that $\mathbb E_\pi[g(X_1)]<\infty$. I am curious about what we can say about $\mathbb E_\pi[g(S_n/n)]$,…
Bravo
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Neural Networks: How to get the gradient vector for the xOr problem?

I'm reading about neural networks, but the material I find is sometimes very abstract or just copies of something. Well, when considering the $xOr$ problem, I have a network in the following structure Which mathematically can be represented as…
David
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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 $\int_0^\infty f(t)\, \mathrm{d}t$ converges. Prove that…
Joel Tan
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Why is the formula for the density of a transformed random variable expressed in terms of the derivative of the inverse?

In this very nice answer, the intuitive explanation of the formula for the density of a transformed random variable, $Y = g(X)$, leads naturally to an expression like $$f_Y(y) = \frac{f_X(g^{-1}(y))}{g'(g^{-1}(y))},$$ where $f_X(x)$ is the density…
R Hahn
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Differentiating $ (y-X\beta)^T(y - X \beta) $ with respect to $\beta$

How do I differentiate $$ (y-X\beta)^T(y - X \beta) $$ with respect to $\beta$. The result I saw was $$X^T(y - X\beta)$$
EA Lehn
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