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Questions tagged [integral]

For on-topic question related to uses of the mathematical concept of an integral, i.e. $\int_a^b f(x)\; dx$. Purely mathematical questions about integrals are better asked at math SE: https://math.stackexchange.com/

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Bounds of integration for the Wishart density

I once took a course that included zillions of exercises concerning the Wishart distribution, but as far as I recall, never mentioned the Wishart density. I asked something about that in this question,...
Michael Hardy's user avatar
3 votes
1 answer
33 views

Understanding how to evaluate the integral causal-effect expression

I have this expression $$ p( Y \mid \text{do}(Z=z)) = \int_{B, S, W, X} dBdSdWdX \ \ P(B | S) P(W | B, S) P(X | B, S, Z=z) \left[ \int_{Z'} dZ' P(Z'| B,S,W) P(Y | B, S, W, X, Z') P(S) \right] $$ ...
Astrid's user avatar
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0 answers
36 views

Does the negative multinomial distribution have a defined CDF?

I have a process that receives categorically distributed random inputs of 5 different types. The 5th type is considered "bad". If the process receives 4 bad inputs before receiving at least ...
ZPears's user avatar
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5 votes
2 answers
321 views

Expected loss function from bias variance trade off (integral help)

I have a hard time understanding this formula. It's from bias-variance trade-off proof. and the expected loss function is as follows: $$L(\hat f) := \mathbb E_D\mathbb E_{(x,y)}[(y-\hat f(x))^2]=\...
Taewooo Kim's user avatar
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24 views

What is the pdf of the integral of a gaussian process and of the ratio of two gaussian variables?

I need to evaluate the moment functions of a zero mean gaussian process that constitutes the mathematical model of the seismic ground acceleration during an earthquake.
Adrian Daniliuc's user avatar
0 votes
0 answers
51 views

Find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$

Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain. We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$...
Anweshan Goswami's user avatar
7 votes
1 answer
96 views

Recurrence formula for the moments of a half-gaussian distribution (on R+)

I am trying to compute an integral that looks like the moments of a Gaussian $\mathcal{N}(\mu, \sigma^2)$, but the main difference is that we only integrate over R+ and not R. I believe we could call ...
Julia Linhart's user avatar
0 votes
0 answers
46 views

Why use integrals when building exponential composite functions?

I recently read this paper, which describes a generalisation for statistical exponential decay models used in ecology. Essentially, the parameter $k$ of the exponential decay function $f(x) = ce^{-kx}$...
Luka Seamus Wright's user avatar
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0 answers
23 views

Monte Carlo Approximation on integral of Gaussian pdf on Convex Domain

I have hard time on estimating the following integral on convex domain ($\mathcal D$) using Monte-Carlo approximation. $$a = \int_{\mathcal D} dx f(x;\mu,\Sigma) $$ where $x \in \mathbb R^d$ and $f$ ...
Interception's user avatar
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Question on the proof step in the theorem 1 of the Gap statistic paper

From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422), $\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
kurtkim's user avatar
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0 answers
36 views

Closed form expression of Bayes Factor

Setting Let data $\sim N(\mu, \sigma^2)$ where we consider $\sigma^2$ known. Prior: $\mu \sim N(m_0, \sigma^2 / n_0)$. Question I'm trying to calculate $A / B$, where $A = p(data | \mu = m_0, \...
user7064's user avatar
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0 answers
34 views

Converting an integral into a probability of some event

Suppose that $X_1, X_2, .....X_n$ are iid random variables from some continuous distribution $F$. Show that $$\int_0^{\infty}(1-F(s+t))f(s)ds=\mathbb{P}(X_1>X_2+t, X_2>0)$$ $$$$Consider the ...
user671269's user avatar
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0 answers
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Highest posterior density [duplicate]

How is to find the value of $k$ for the following equation which gives $100(1-\alpha)\%$ highest posterior density? $$\int_{\theta:\pi(\theta|\mathbf x)>k}\pi(\theta|\mathbf x)d\theta=1-\alpha,$$ ...
user149054's user avatar
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0 answers
18 views

Solving integrals using incomplete gamma function (upper gamma rule)

I am attempting to integrate this function, ∫15x^(0.28) * e^(-0.21x) dx and am struggling with what techniques to apply. The lower boundary is 0 and the upper boundary is infinity. From research, I ...
user avatar
0 votes
1 answer
73 views

Any closed-form solution to this integral (multivariate exponential)?

Here is the probability density function (unnormalized) of a covariance matrix: (from a Bayesian perspective): $$ f(\boldsymbol{V})\propto \det(\boldsymbol{V})^{-\frac{N+J+1}{2}}\int_{\mathbb{R}^{K}}\...
fan455's user avatar
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0 answers
22 views

Computing an integral that reduces to $\mathbb{P}[X>Y]$

Problem Evaluate $$I=\int_{-\infty}^\infty \frac{e^{-\frac{1}{2}\left(\frac{x-\mu)}{\sigma} \right)^2}}{\sigma \sqrt{2 \pi}}\frac{1}{1+e^{-x}}\, \mathrm{d}x$$ My attempt Now the first part of the ...
Bhoris Dhanjal's user avatar
0 votes
0 answers
49 views

Probability that a random variable has a greater absolute value than the sample mean of iid random variable

I have encountered a problem which involves finding the probability that an observation of a random variable has a greater absolute value than the sample mean of an independent set of observations of ...
chasmani's user avatar
  • 165
1 vote
1 answer
62 views

Best way or rules of thumbs for evaluating 1d integrals from randomly sampled grid points

I have a 1D domain (let's say the interval $(0,1)$) on which I randomly sample $N$ points from the uniform distribution. I have a function $f\colon (0,1) \to \mathbb{R}$ which is integrable. What is ...
math_guy's user avatar
  • 111
0 votes
1 answer
78 views

Marginal Gaussian distribution

I have a confusion over an integral involving a multivariate and a univariate Gaussian. We know that in the case of two multivariate Gaussians the following is true: $$ \int \mathcal{N}(\mathbf{y}|\...
ngiann's user avatar
  • 1,249
0 votes
0 answers
17 views

Marginal Filtering and Smoothing Distribution

I want to know the calculation of marginal filtering distribution and smoothing distribution from the joint distribution. For example, Joint filtering distribution: $$ p\left(x_{1: t} \mid y_{1: t}\...
stander Qiu's user avatar
0 votes
0 answers
123 views

Integral of multivariate gaussian

I'm calculating a posteriori distribution, which is solving this integral: $$ \int d^Jx \frac{\exp \left[\left( \vec{x}^\intercal \Omega^{-1} \vec{x} \right)/2\right]}{(2\pi)^{J/2}\sqrt{|\Omega|}} \...
chuse's user avatar
  • 780
1 vote
2 answers
53 views

How to obtain the following marginal distribution

I'm struggling on how to calculate the marginal $f(x)$. I'm trying to integrate $f(x,y)$ by $y$, but I couldn't solve it. The pdf is the following. $\displaystyle f(x) = \int_0^{\infty} f(x,y)\text{d}...
Ga13's user avatar
  • 280
0 votes
1 answer
105 views

Solve equation for a given set of parameters

I'm modelling a process for which the probability of the event (stop) not happening before time $t$ is $e^{-\lambda\cdot t }|\lambda>0$. When the event happens, the process stops running for a ...
Jon Nagra's user avatar
  • 313
2 votes
0 answers
21 views

Showing stationarity thanks to integrals

Let $X_t$ be a stationary process with spectral representation $$X_t = \int_{\left[-\frac12 , \frac 12\right]}e^{2 i \pi f t}\,dZ(f).$$ Assume that $E\lvert dZ(f)\lvert^2=\lvert G(f) \rvert^2\,dF(f)$ ...
Kilkik's user avatar
  • 355
4 votes
1 answer
233 views

Rewriting the expectation of f(x) by means of its derivative

I have a question regarding this proposition. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an a.e. differentiable function so that $\int \frac{\left|f^{\prime}(x)\right|}{(1+|x|)^s} d x<\infty$ ($...
Eryna's user avatar
  • 309
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0 answers
56 views

How to numerically get expectation of a non-linear function of a normally distributed random variable

I'm trying to calculate the expectation of the following numerically: $$\mathbb{E}[V(\theta)]$$ where $\theta\sim N(\mu,\sigma^2)$ and $V(\theta)$ is strictly increasing. I'm struggling to understand ...
Anonymouslylost's user avatar
5 votes
1 answer
171 views

How to deduce $ \mathbb{E}(\sqrt{X}) < \infty \implies\int_{\mathbb{R}^+} (1 - F(x))^2 dx < \infty,~X$ being a non-negative integrable rv?

Let $X$ be non-negative random variable and $F$ be its distribution function. Prove the following implications: $$ \mathbb{E}(X) < \infty \Longrightarrow \mathbb{E}(\sqrt{X}) < \infty \...
Thành Nguyễn's user avatar
1 vote
0 answers
47 views

Expectation of a Compound Poisson Distribution

I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula. I would like to know how to prove this: $$E[X]=\int x e^{-v(\...
Enrico's user avatar
  • 211
2 votes
1 answer
41 views

Numerical integration over a Random Forest Regression Model

I am trying to compare the accuracy of a polynomial model and a Random forest regression model in predicting a variable Y but also the integral of this variable , With the polynomial model, it is ...
Issamyax's user avatar
1 vote
1 answer
63 views

How could I evaluate $A = \int_0^1 \log\left(\theta^s(1-\theta)^{n-s}\right)p(\theta)d\theta$?

Suppose that I want to evaluate the following integral: $$A = \int_0^1 \log\left(\theta^s(1-\theta)^{n-s}\right)p(\theta)d\theta,$$ where $p(\theta)\equiv$ Beta$(ws+1, w(n-s)+1)$ and $n$, $w$, and $s$ ...
Ron Snow's user avatar
  • 2,179
0 votes
0 answers
47 views

Expectation of squared integral over distribution

Right now, I have some function $g(x,\theta)$ where the expectation this function at a given $x$ evaluated over $\theta$ $E_\theta[g(x,\theta)]$ is known, and I want to upper bound (or compute if at ...
George's user avatar
  • 131
2 votes
0 answers
69 views

Why are Square Integrable Functions important in Statistics?

I'm reading a paper by Giles Hooker on Functional Decomposition through the use of Functional ANOVA. In the paper he defines a function: $$ F(x) : \mathbb{R}^k \rightarrow \mathbb{R} $$ and explicitly ...
Connor's user avatar
  • 635
1 vote
1 answer
115 views

Derivative of an integral of a random normal variable

Let $x$ be a random normal variable with pdf $h(x)$ and CDF $H(x)$. Also let $\alpha$ be a constant, and $x^\star$ a variable. I am trying to take the following derivative: $$\frac{d}{d x^\star} \bigg ...
phdstudent's user avatar
1 vote
0 answers
40 views

Find marginal distributions of $f(x,y) = 1$ within shaded area A, given by boundaries $y = x^2, ~y = x^2 + 1$ and $ 0 < x < 1.$

I understand that I'm supposed to integrate with respect to x and y to find the marginal distribution of Y and X respectively, but my answers seem wrong. I got marginal distribution of $X = 1$ and of $...
inightyDAb's user avatar
1 vote
1 answer
57 views

How to combine two integrals containing the PDFs of a variable and its linear transform?

Original Post: Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$. I want to compute $Z(x) = F(x) - G(y)$. What I have ...
Philipp's user avatar
  • 13
3 votes
1 answer
128 views

Uniform ergodicity of a Gibbs sampler

We consider a classical data set from Gelfand and Smith containing the information about ten nuclear power plant pump failures. We are interested in the failure intensity of each pump and we employ ...
Alexandria Cortez's user avatar
0 votes
1 answer
57 views

Eliciting a Gamma informative prior in a Gamma–Poisson Bayesian problem

I employ the Gamma–Poisson conjugate family for my statistical model. I want to use an informative prior. From theory, I know that the values of the Gamma-distributed random variable lie within the ...
Valerio's user avatar
  • 37
1 vote
1 answer
251 views

Density from characteristic function: Durrett example 3.3.8 and 3.3.9

Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function $$f(y) = \frac{1}{2\pi} \...
Phil's user avatar
  • 626
6 votes
2 answers
568 views

Applying Leibniz's integral rule to the Gaussian distribution's normalization condition

I'm working on problem 1.8 of Bishop's Pattern Recognition and Machine Learning and am having a hard time understanding one of the technical details in a solution that I found online. Specifically, ...
SayNo2Decaf's user avatar
1 vote
1 answer
152 views

Integral w.r.t. a cumulative density function

I was reading a paper and couldn't understand the following transition. Could someone tell me where the term of $p^k (\frac{1}{2} − c′)$ comes from in the following transition? Def: Cumulative ...
Rowing0914's user avatar
0 votes
1 answer
156 views

Derivation of the LogNormal CDF from PDF [duplicate]

I've been trying to derive the CDF of the lognormal distribution. I got this far but now I'm stuck. $F(x) = \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x\frac{1}{z}e^{-t^2}dz$ where $t = \frac{\ln(z)-\...
Toilet Paper's user avatar
2 votes
1 answer
152 views

Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian

Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$. I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
tcengel's user avatar
  • 23
1 vote
0 answers
63 views

Differential entropy / variance

So I‘m basically more or less trying to prove what is stated in the answer by syeh_106 here: Is differential entropy always less than infinity? if the variance is finite, then the differential ...
Parinn's user avatar
  • 83
0 votes
0 answers
80 views

Confidence interval for evaluating an integral via Monte-Carlo sampling

I am trying to evaluate the following integral using Monte-Carlo: $$ \langle f \rangle = \int \mathrm{d}x~ \rho(x) f(x) $$ where $\rho(x)$ is a normalized positive function. The integration is ...
Yifan Lai's user avatar
6 votes
1 answer
379 views

Compute median of continuous distribution using integrate() in R

sorry if this may sound very silly or stupid, but I'm following a stats course in my uni and I have some exercises in which I have to compute the median of a continuous distribution. I know that, if ...
norberto's user avatar
0 votes
0 answers
57 views

How to perform calculations when the integral of a mixed exponential distribution pdf does not give 1?

Let Y be the time, for a mixture distribution with two exponential distributions, each multiplied by a and b and having 2 different parameters. How can calculations such as mean, variance and ...
tcengel's user avatar
  • 23
3 votes
1 answer
84 views

From density function to cumulative distribution function?

Given $$f(y)=\theta/(\delta^{\theta}y^{\theta+1})\mathbb{1(y>1/\delta)}$$ where the last factor is the indicator function, and I am asked to compute the Cumulative Distribution Function of y: $$\...
pommefatale's user avatar
1 vote
0 answers
45 views

Is there anyway to calculate the integral of a trace? [closed]

I would like to calculate the integral of a scalar function as follows: $$f(x)=\mathrm{tr}((\mathbf{A}x+\mathbf{B})^{-1}\mathbf{B}),$$ where $\mathbf{A}$ and $\mathbf{B}$ are two $n\times n$ positive-...
Deku's user avatar
  • 21
1 vote
0 answers
136 views

Marginal density of dirichlet distribution

I'm studying BRML. In this book, a Dirichlet distribution is defined as $$ p(\alpha | u) = \frac{\Gamma(\sum_{q=1}^Q u_q)}{\prod_{q=1}^Q \Gamma(u_q)} \delta_0 \left( \sum_{q=1}^{Q} \alpha_q - 1 \right)...
yeomjy's user avatar
  • 11
1 vote
0 answers
124 views

Compute Expectation of Product Using Joint Survival Function

I know that, for a nonnegative random variable $X$, $$ E[X] = \int x dF(x) = \int S(x) dx$$ where $F(x)$ and $S(x)$ are the CDF and survival function of $X$, respectively. This was derived using ...
Peter_Pan's user avatar
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