Questions tagged [calculus]

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How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this. ...
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28 views

Calculus with quadratic B-spline

After fitting a quadratic B-spline in R with the cobs package : Rbs <- cobs(x,y, constraint= "decrease", pointwise = con) I would like to do some calculus on ...
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what is the relation between change of variables in probability and calculus?

I want to know if the change of variables formula in probability is a special case of the change of variables in calculus, or something different. Trying to think for myself, first write the CoV for ...
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1answer
31 views

Symmetry between integrals including absolute value

So I came across below symmetry in my probability course that I can't understand. I understand how the lower bound changes when removing the absolute value operator, but how does the 2 disappear?
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A maximization problem involving random variables: a special case

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \color{blue}0 \\ \mu_Y \end{pmatrix} , \...
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25 views

A maximization problem involving random variables

Consider random variables $X$ and $Y$ that are jointly normally distributed, $$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim \mathcal{N} \left[ \begin{pmatrix} \mu_X \\ \mu_Y \end{pmatrix} , \begin{...
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How to calculate logarithmized average monthly returns

I am currently reading the following paper https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2750064 . On page 57 the researchers state that they calculate the logarithmized average monthly returns ...
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1answer
66 views

Find marginal distribution

The random vector $(X,Y)$ is uniformly distributed over $$D=\{(x,y): 0 \leq x \leq 2 , 0 \leq y \leq 2-x\}.$$ Find the marginal distribution of the random variables $X$ and $Y$. For the radom vector $...
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1answer
33 views

Derivative of expectation where the variable appears in the integration limit and in the integrand?

I want to calculate the derivative of $$\varphi(\mu) = \int_{-\infty}^{\mu} r(x-\mu) f(x)dx,$$ wrt to $\mu$, where $r$ is a function and $f$ is a density function. How can I account for the presence ...
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122 views

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 ...
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149 views

Solving integrals in R

I would like to write an R function for solving the following equation: Essentially I would like to be able to set or vary the parameters values of "m" and "s" and those parameters in "p(t)" ...
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53 views

Conditional Density Estimate Loss. Why the double integral?

I read RFCDE: Random Forests for Conditional Density Estimation. Just like it sounds, these folks trained random forests for making conditional density estimates. At inference time, density estimates ...
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560 views

How to manually calculate odds ratio for continuous variables?

In school, long before learning about logistic models, I've been taught how to calculate odds ratios by hand. Formula was based on a contingency table, just like this: This is very easy to ...
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80 views

Properties of Kernel Density Estimators

Given Let $X \in \mathbb{R}$ be a real-valued random variable with theoretical probability density function (pdf) $f(x)$ and corresponding cumulative distribution function (cdf) $F(x)$. Let $X_1, X_2,...
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Threshold optimization with two parameters where first parameter need to be minimum and second value to be maximum

We have threshold values ranging from 1 to 10 where attributes p1, p2 increase with increase in threshold. Our intention is to find threshold with minimum p1 and maximum p2. It would be of great help ...
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212 views

Law of Iterated Expectations Example

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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25 views

Expectation / Summation inequality with multiple indices

Part of a solution to an exercise in the book Stochastic Processes From Application to Theory (exercise 87) is the following summations: $$\sum_{i\ge1}P\left(I_i \ge i \right) = \sum_{j \ge i\ge1}P\...
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270 views

Relationship between first and second order condition of convexity

Suppose we have a function $f(\boldsymbol{x})$ and its hessian, i.e $\nabla_{\boldsymbol{x}}^2f(\boldsymbol{x})$, equals $\mathbf{0}$. We know that for convexity $\nabla_{\boldsymbol{x}}^2f(\...
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85 views

What does gradient with respect to a function mean [duplicate]

I am trying to understand this paper better Greedy function approximation: A gradient boosting machine, but I start having difficulty at around Equation (6) and (7). What does a gradient w.s.t. to an ...
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1answer
40 views

Simple Appplication of Law of Iterated Expectation

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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0answers
24 views

Verifying sub-exponential property when a random variable is not sub-gaussian pro

I am referring to the Example 2.4 (page 16) in this book chapter https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf Suppose $Z \sim N(0,1)$ and random variable $X=Z^2$. ...
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1answer
136 views

Expected Loss Calculation

How do you solve the integral $$E(L(\theta_A,\theta_B)) = \int_0^1\int_{\theta_B}^1(\theta_A - \theta_B)f(\theta_A)f(\theta_B)d\theta_Ad\theta_B$$ where $\theta_A \sim Beta(\alpha_1, \beta_1)$ and $\...
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54 views

Simple Log-likelihood question

I've got a simple question about deriving log-likelihoods. I am stumped by the following--> If the log-likelihood is: 𝑙(𝜆1,𝜆2) = 𝑦1 log(𝜆1𝐹1)−𝜆1𝐹1 −log((𝑦1)!)+𝑦2 log(𝜆2𝐹2) −𝜆2𝐹2 −log⁡ (...
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17 views

Maximize profits by setting price

I am working on a model to set a price that maximizes profits. The equation for profits is: Profits=price x (# sold) - (fixed cost) x (# sold) I have models ...
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1answer
2k views

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 ...
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47 views

Can Regularization by achieved using Relative Sensitivity?

In a Mathematical Model we measure the sensitivity of the output with respect to the parameters and it is desirable that a small change in a parameter doesn't lead to wild fluctuations in the output ...
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1answer
3k views

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{...
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1answer
629 views

gradient descent momentum vs step size

In the gradient descent method, the learning rate (which is multiplied by the results of the gradient on each weight) identifies the size of the step (steep down) that the algorithm takes in each ...
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1answer
370 views

Help with derivation of Mean Field Variational Inference

I am studying Variational Inference using Bishop's book: Pattern Recognition and Machine Learning. At the moment, I am struggling to understand the Lower Bound derivation for the Mean-Field ...
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1answer
344 views

Conditional expectation student's t distribution

I was looking at the student's t distribution and was interested in the following conditional expectation: $$ E[X|X\geq t_v^{-1}(\alpha)] = \frac{g_v(t_v^{-1}(\alpha)}{1-\alpha}\left( \frac{v + (t_v^{-...
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3k views

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 ...
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2answers
312 views

deriving z score for 95% confidence interval [duplicate]

95% of the area under the standard normal distribution lies within 1.96 standard deviations away from the mean (0). This 1.96 number is used to construct 95% confidence intervals. I was just ...
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1answer
434 views

Calculating t-SNE gradient (a mistake in the original t-SNE paper)

This is specific to the way the gradient of the KL divergence Loss function was derived in the original paper Visualizing Data using tSNE. In the Appendix A (Page 21), where they derive the gradient, ...
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1answer
663 views

How to differentiate the distribution function of lognormal distribution with respect to its parameters?

How to differentiate the distribution function of lognormal distribution with respect to its parameters? What solution will we get? I know if differentiate wrt variable, we will get density function.!
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77 views

Transformation of random variables in Generative Adversarial Nets

Let $x$ and $z$ be real-valued random vectors, where $x = g(z)$, and $g$ is an invertible, continuous and differentiable transformation of $z$. Then $p_z(z) = p_x(g(z)) \lvert det(\frac{\partial g(z)}{...
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2answers
3k views

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 ...
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1answer
27 views

What is the right method to estimate rate of change in daily values over a period of time?

I would like to ask you what is the best (or the right way) to calculate the rate of change in air temperature over a period of time and then see how this rate changes over time. Daily air temperature ...
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29 views

Re-express the lognormal density in terms of its mean value

I wonder if it is possible to rewrite the lognormal density function, $$f(x)=\frac 1 {x\sigma\sqrt{2\pi}}\exp\bigg(-\frac{(\log x-\mu)^2}{2\sigma^2}\bigg)$$ so that the mean value, $\exp(\mu+\frac{ \...
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1answer
67 views

Seeing chi square minimization problem as calculus of variation problem

My question will be less informative because right now I myself don't know where I am heading. Basically I want to know that whether it is possible to think a chi-square minimization problem as a ...
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0answers
558 views

Meaning of Jacobian of the transformation for pdf of function of random vectors

I am studying multivariate statistics and I don't understand the meaning of Jacobian of the transformation for pdf of function of random vectors. If I have a random vector, let's say bivariate, (X,Y)...
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1answer
187 views

What is the intuition behind neural networks?

Everywhere in the theory of neural networks, authors saying that idea came about by observing the work of the human brain. But I can not believe in that. I guess, everything is much simpler and neural ...
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1answer
255 views

Is there local minimums in MSE function?

Here is "mean squared error" function: C = $\frac{1}{2n}$ * $\sum(length(y - a)^2)$ As I understand, this is like paraboloid in multidimensional space. So, I guess, there is only one extremum: global ...
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1answer
390 views

What is the second derivative of a B-spline?

A B-spline of degree $j$ is defined at knots $\vec k$ by the Cox-de Boor recursion formula \begin{align} B_{i,1}(x) &= \left\{ \begin{matrix} 1 & \mathrm{if} \quad ...
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1answer
106 views

Finding integral of minimum function. yn(1-y)^(n-1)

This is for a homework question in which I am trying to find the $E(Y_n = min\{X_1,...,X_n\}$). So far I have found that the minimum cdf, as below. The minimum of $X_i$ is when all of $X_i > x$. ...
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141 views

Solution verification - calculation of second derivatives of multinomial probit log-likelihood function

The initial function of log-likelihood of multinomial probit model with $J$ alternatives: $ ln \ell=\sum_{i=1}^N\sum_{j=1}^{J-1} y_{ij} \cdot ln \Phi(\sum_{k=1}^Kx_{ik}\beta_{kj})+ ({n_i-\sum_{j=1}^{...
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1answer
1k views

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)...
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1answer
2k views

Why is optimisation solved with gradient descent rather than with an analytical solution? [duplicate]

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 ...