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/
382
questions
5
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1
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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:
$$
\mathbb{E}(X) < \infty \Longrightarrow \mathbb{E}(\sqrt{X}) < \infty \...
- 185
1
vote
0
answers
32
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Expectation of a Compound Poisson Distribution
I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula.
I would like to know how to prove this:
$$E[X]=\int x e^{-v(\...
- 201
0
votes
0
answers
30
views
Confused with notations about random variable and expectation
I asked a question originally here, but my notations were confusing and I couldn't convey properly in terms of statistics. Notations in statistics is a bit new to me, because my background is mostly ...
- 81
1
vote
1
answer
21
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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 ...
- 11
0
votes
1
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54
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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 $s$ ...
- 1,799
0
votes
0
answers
24
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Integral of normal likelihood and multivariate normal prior
I'm updating cluster assignments in the context of a non-parametric Bayesian mixture model. When computing the probability of starting a new cluster, in the absence of cluster parameters (and using a ...
- 73
0
votes
0
answers
33
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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 ...
- 111
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0
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43
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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 ...
- 349
0
votes
1
answer
63
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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:
$$\frac{d}{d x^\star} \bigg ...
- 105
1
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0
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Find marginal distributions of $f(x,y) = 1$ within shaded area A, given by boundaries $y = x^2, ~y = x^2 + 1$ and $ 0 < x < 1.$
I understand that I'm supposed to integrate with respect to x and y to find the marginal distribution of Y and X respectively, but my answers seem wrong. I got marginal distribution of $X = 1$ and of $...
- 11
1
vote
1
answer
54
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How to combine two integrals containing the PDFs of a variable and its linear transform?
Original Post:
Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.
I want to compute $Z(x) = F(x) - G(y)$.
What I have ...
- 13
3
votes
1
answer
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Uniform ergodicity of a Gibbs sampler
We consider a classical data set from Gelfand and Smith containing the information about ten nuclear power plant pump failures. We are interested in the failure intensity of each pump and we employ ...
0
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1
answer
43
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Eliciting a Gamma informative prior in a Gamma–Poisson Bayesian problem
I employ the Gamma–Poisson conjugate family for my statistical model.
I want to use an informative prior.
From theory, I know that the values of the Gamma-distributed random variable lie within the ...
- 19
0
votes
1
answer
93
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Density from characteristic function: Durrett example 3.3.8 and 3.3.9
Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function
$$f(y) = \frac{1}{2\pi} \...
- 562
6
votes
2
answers
434
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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, ...
- 63
1
vote
1
answer
71
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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 ...
- 153
0
votes
1
answer
87
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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)-\...
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2
votes
1
answer
108
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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: $$ f_{r,\theta}(r,\...
- 23
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votes
0
answers
40
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Substitution principle - Sample mean estimator
We can read in Keith Knight - Mathematical Statistics Example 4.22 the following.
EXAMPLE 4.22: Suppose that $\theta(F)=\int_{-\infty}^{\infty} h(x) d F(x)$. Substituting $\widehat{F}$ for $F$, we get
...
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1
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0
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36
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Differential entropy / variance
So I‘m basically more or less trying to prove what is stated in the answer by syeh_106 here:
Is differential entropy always less than infinity?
if the variance is finite, then the differential ...
- 83
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0
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22
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Numerical Solution of two convoluted stable paretian random variables
I am trying to numerically compute the joint density of X and Y, where both are stable paretian distributed random variables with different alphas (1.4 and 1.7).
I can compute the PDF via inversion ...
0
votes
0
answers
45
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Confidence interval for evaluating an integral via Monte-Carlo sampling
I am trying to evaluate the following integral using Monte-Carlo:
$$
\langle f \rangle = \int \mathrm{d}x~ \rho(x) f(x)
$$
where $\rho(x)$ is a normalized positive function. The integration is ...
6
votes
1
answer
261
views
Compute median of continuous distribution using integrate() in R
sorry if this may sound very silly or stupid, but I'm following a stats course in my uni and I have some exercises in which I have to compute the median of a continuous distribution.
I know that, if ...
- 73
0
votes
0
answers
51
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How to perform calculations when the integral of a mixed exponential distribution pdf does not give 1?
Let Y be the time, for a mixture distribution with two exponential distributions, each multiplied by a and b and having 2 different parameters. How can calculations such as mean, variance and ...
- 23
3
votes
1
answer
57
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From density function to cumulative distribution function?
Given $$f(y)=\theta/(\delta^{\theta}y^{\theta+1})\mathbb{1(y>1/\delta)}$$ where the last factor is the indicator function, and I am asked to compute the Cumulative Distribution Function of y:
$$\...
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0
answers
22
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Clarification on marginalizing out variables in conditional probability statements
If I have the joint distribution ($\bar{y}$ denotes a vector and $\bar{y}_{1:K}$ says there are K of them):
$$P(\theta_{1:K}, \bar{y}_{1:K}, \omega | X_{1:K})$$
and this joint distribution factors ...
- 320
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0
answers
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Does the following "approach" for integrating over the data space makes sense?
Suppose that we have the following posterior and prior distributions
$p(\mu|x,m_{1}(x),s_{1}(x)) = Normal(\mu;m_{1}(x),s_{1}(x))$ and $p(\mu|m_{2},s_{2})$
The $m_{1}(x),s_{1}(x)$ indicate that the ...
- 2,070
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vote
0
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43
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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-...
- 21
1
vote
0
answers
88
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Marginal density of dirichlet distribution
I'm studying BRML.
In this book, a Dirichlet distribution is defined as
$$
p(\alpha | u) = \frac{\Gamma(\sum_{q=1}^Q u_q)}{\prod_{q=1}^Q \Gamma(u_q)} \delta_0 \left( \sum_{q=1}^{Q} \alpha_q - 1 \right)...
- 11
1
vote
0
answers
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Compute Expectation of Product Using Joint Survival Function
I know that, for a nonnegative random variable $X$,
$$ E[X] = \int x dF(x) = \int S(x) dx$$
where $F(x)$ and $S(x)$ are the CDF and survival function of $X$, respectively.
This was derived using ...
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votes
0
answers
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Improper Prior in Logit and Probit Models: Proper Posterior Conditions
Let
$y_i \vert p_i \sim \mathrm{Bernoulli}(p_i)$,
$p_i = F_h(X_i^\prime \beta) \ \ , \ \ h = 1,2 \ ,\ \ X , \beta \in \mathbb R^p$,
where $F_1(x) = (2\pi)^{-1/2}\int_{-\infty}^x \exp(-t^2/2) \ dt \ $ ...
- 31
1
vote
0
answers
31
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How to get to the formula of this joint probability
In Bishops book, "Statistical Pattern Recognition", in Chapter 8 about Graphical models he introduces the following 3-node graph:
The goal is to investigate the independence properties of ...
- 465
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votes
0
answers
14
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Integral over subset of parameter space in which densities of parameters limit to 0
Let $\Theta \subseteq \mathbb{R}$ denote the space of a parameter $\theta$. Also define the subsets $\Theta^* \equiv \left\{\theta: c - \varepsilon < c < c + \varepsilon\right\}$ and $\Theta^+ \...
- 63
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votes
0
answers
38
views
Expected value of a complicated function
I want a closed form/ semi closed form of the expected value of a complicated function.
The function looks like this,
$$ f(x) = \frac{\sin(A \frac{x}{2})}{\sin(\frac{x}{2})} \frac{\sin(M B\frac{x}{2})}...
- 81
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votes
0
answers
18
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Improper integrals of symmetric functions [duplicate]
First time poster here, so I apologize for any formatting errors. I recently came across the improper integral ∫xdx from -∞ to ∞ and have had a hard time understanding why it isn't zero. My approach ...
2
votes
0
answers
39
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integral related to a general bivariate copula C(u,v) of |u-v|
I'm trying to compute the following integral over the unit square $I^2=[0,1]^2$:
$$
\int_0^1\int_0^1 |u-v|dC(u,v),
$$
where $C(u,v)$ is a generic bivariate copula, which should be equal to
$$
1-2\...
- 31
0
votes
0
answers
15
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Mollifying Wiener-Khinchin integral to agree with observed autocorrelation
I'm looking for a way to characterize the autocorrelation of non-stationary processes. Obviously the Wiener-Khinchin theorem does not hold, so the task might be hopeless, but nonetheless there might ...
- 215
3
votes
1
answer
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Is there a meaning to the integral of $x \times f(x)$ over a range that is not infinite?
I know that the expected value can be computed as :
$\mathbb{E}(X) = \int_{-\infty}^{\infty}xf(x)dx$
What if we do not do the integral over the whole range but only up to some value? Would there be ...
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1
vote
0
answers
18
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Solving integral with generic pdf [closed]
I have a continuously differentiable function $h:[\underline{x},\overline{x}] \rightarrow \mathbb{R}$, and a continuous random variable $X$ distributed according to a cdf $F$, with full support on $[\...
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2
votes
1
answer
36
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Bayesian statistics: what is the variable we are integrating in?
This is a screenshot from Bayesian Data Analysis by Gelman.
I am a little bit confused by Equation 1.4 (first and second lines), having read Equation 1.3.
In Equation 1.3, the variable of integration ...
- 8,547
2
votes
2
answers
659
views
transformation of a kernel density estimate to uniform distribution
I am interested in estimating the expected value of a function, $f(x)$ with respect to a probability density function, $P(x)$.
I am exploring a method that requires I change variables from the ...
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1
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0
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For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$
Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$.
My attempt (using integration by parts):
\begin{align}
\int_0^k y \,dF(y) &\leq E(Y) \...
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votes
1
answer
328
views
Multivariate Gaussian probability mass inside a sphere
Assume I have some d-dimensional multivariate gaussian $X\sim\mathcal{N}\left(\mu,\Sigma\right)$
and some sphere $C=\left\{ x:\left\Vert x-z\right\Vert_2\le r\right\}\subseteq\mathbb{R}^{d}$.
I was ...
- 1
1
vote
1
answer
695
views
Finding E(XY) for joint probability density
$Joint \:probability\;f(x,y) = 2/3 \:for\: 0 < x < 1, 0 < y < 2, x < y, and\: 0\: otherwise $
$E(XY)=\int_{0}^{1}\int_{x}^{2} \frac{2}{3}xy \:dy \:dx = \frac{7}{12} - (1)$
$E(XY)=\...
- 45
0
votes
1
answer
97
views
Solve an inequality finding the upper bound
Suppose that there exists a constant $C$ such that the following relation
holds for all $G$:
\begin{equation*}
\vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert
\end{equation*}
Suppose that ...
- 57
2
votes
0
answers
425
views
Expected value of the largest order statistic for $Uniform(\theta,2\theta)$ [duplicate]
I'm struggling to find when $X_1,\ldots,X_n \sim Uniform(\theta,2\theta)$, how the expected value of the largest order statistic is $E[X_{(n)}]=\dfrac{2n+1}{n+1}\theta$. I can find that the density of ...
- 3,505
8
votes
2
answers
884
views
Probability of collision: mathematical vs probabilistic modeling
$\newcommand{\icol}[1]{% inline column vector
\left(\begin{smallmatrix}#1\end{smallmatrix}\right)%
}$
Scenario:
Let's consider a road segment on which there is continuous flow of cars circulating at ...
- 153
24
votes
5
answers
1k
views
How to show that this integral of the normal distribution is finite?
Numerically, I have noticed that
$$\int_{-\infty}^{\infty} \dfrac{\phi(x)^2}{\Phi(x)}dx < \infty$$
where $\phi$ and $\Phi$ are the standard normal pdf and cdf. However, I do not see how to prove it....
- 241
0
votes
0
answers
128
views
integration of product of a gaussian pdf and a student-t pdf
I want to perform the following integration wrt $x$:
$$\int_{-\infty}^{\infty}\frac{1}{\sqrt(2\pi\sigma^2)}e^(\frac{-(y-hx)^2}{2\sigma^2})[(1+\frac{x^2}{b})^{-(\frac{b+1}{2})}]dx$$
Here first part is ...
- 1
1
vote
0
answers
21
views
HOW to determine $S$ in $SEIR$ epidemiology model
With the contact rate parameters for a $SEIR$ model as follows:
$$
\begin{aligned}
\beta &= 0.139\\
\gamma &= \frac{1}{10.4}\\
\sigma &= 0.1\\
R_0 &= 1.4456\\
N &= 3,787,000\\
I(0) ...
