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1
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0answers
21 views

Monte carlo integrations with metropolis hastings step

Consider the following problem: Suppose we want to compute the following integral $$ f(y_1|y_2) = \int_{\theta} \int_{x_1} \int_{x_2} f(y_1| x_1,x_2,\theta, y_2) f(x_1,x_2|\theta,y_2) f(\theta | ...
2
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0answers
77 views

A Integral question

I am having problem with integral: $$\int_{-\infty}^\infty\frac{1}{2\pi} e^{-itx} \left( 0.5 \left( e^{-4t^2} + 1 +\cos(t) + \cos(2t) \right)\right) \,dt$$ @DilipSarwate I have proved lim sup(cf)=1 ...
4
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2answers
35 views

Expectation of von Mises Fisher Distribution

The von Mises- Fisher distribution is defined as $$ \frac{\kappa^{p/2-1}}{2\pi I_{p/2-1}(\kappa)}\exp(\kappa \mu^Tx) $$ It is defined over the unit sphere i.e. $||x||_2^2=1$. My question is what is ...
3
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2answers
69 views

conditional expectations value

I need to calculate the following integral $$\int_{\mu+c}^{\infty} y\cdot \frac{1}{\sigma\sqrt{2\pi}}e^{(y-\mu-w)^2/2\sigma^2}dy$$ So essentially $y\sim N (\mu+w, \sigma^2)$ and im trying to ...
0
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1answer
24 views

Compute the cumulative hazard of a time interval

I'm a little confused in calculating the cumulative hazard within a time interval. We know that $H(t)=\int^{t}_{0}h(u)du$, if I have $\Delta t=t_1-t_0$ (1) $H(\Delta t)=\int^{\Delta t}_{0}h(u)du$ ...
1
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1answer
55 views

Explanation of density rewriting?

Can somebody please explain the math behind this statement to me? I am not sure how they represent the left hand side by that integral and finally how it is proportional to that. \begin{align} ...
1
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1answer
33 views

[Revised]Proving the expected \bold{density} of being the Nth order statistics is decreasing in sample size

(Sorry that I've previously formulated the question in a wrong way, which confused everyone including myself. This is a better version of the question. Thanks!) Here's another order statistics ...
3
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2answers
116 views

Proving some properties of expected first order statistics with respect to sample size

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as $E(\mathcal{O}^n_1)= ...
1
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0answers
63 views

Integrate first derivation - area under curve of first derivation

I want to calculate the area under the first derivate. Can I do that with splinefun()? ...
2
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1answer
55 views

Moment generating function if the PDF is $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$

Let $z$ a random variable with PDF : $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$, where $0\leq z \leq d$, $F$ is the Hypergeometric function, $k$ is a positive integer, $-a+k+1 ...
7
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1answer
113 views

Integrating an empirical CDF

I have an empirical distribution $G(x)$. I calculate it as follows x <- seq(0, 1000, 0.1) g <- ecdf(var1) G <- g(x) I denote $h(x) = dG/dx$, ...
1
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3answers
149 views

Why is the marginal liklihood term(the denominator) in Bayes' updation not a function of theta(parameters)?

The snippet from the book is pasted above. I don't understand how the denominator is not a function of theta as it is integrated over theta. Thank you.
1
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0answers
89 views

Moment of random variable on a integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
1
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0answers
102 views

Monte Carlo integration

I am calculating a simple integral $\int^1_{-2} \exp{x^2}(x+1)dx $ with Monte Carlo method using a linear density function $p_\xi (x) = \frac{4}{9} + \frac{2}{9} x $. Let say I have a a sample which ...
0
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1answer
31 views

Approximation of Pr(X > a), with X a multivariate normal rv

Let $X = (X_1, ..., X_p)$ a random variable with a $N(\mu, \Sigma)$ distribution. $$ $$ $$ \Pr(X_1 > a_1, ..., X_p > a_p) \\ =\int_{a_1}^\infty ... \int_{a_p}^\infty (2\pi)^{-p/2} ...
0
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1answer
59 views

Integration problem in Bayes factor calculation for multinomial model

This is one integration problem I encountered during the calculation of Bayes factor between two models given data $D$ One of the model, $M_0$ assumes the data accords to multinomial distribution, ...
1
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2answers
78 views

Ordered gamma variables led to an ugly integral

Suppose $X_1,X_2,...X_n$ are i. i. d. random variables with p. d. f. $$f(x)=xe^{-x}I_{(0,\infty)}\!(x)$$ and let $Y_1,...,Y_n$ be the order statistics for these variables. a) Find the conditional p. ...
1
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1answer
56 views

Integral of a function with uniform kernel

I am trying to understand question 9-1 on p.334 in Cameron & Trivedi (link) where I have to calculate the bias of a kernel density estimate at $x=1$ and $n=100$, where we assume that the ...
2
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1answer
93 views

How to arrive at a specific formulation of the relative median deviation?

I am an economist currently working with this book: Frank Cowell - Measuring Inequality On page 25 a formulation of the relative mean deviation is given as follows: $$ M = 2 \left[ ...
1
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2answers
50 views

What is the expected partial value function really called?

If f is a pdf, the integral of x*f(x) over the entire range where f(x) > 0 gives, of course, the expected value. Suppose that integrate the same function, x*f(x) from negative infinity up to t, ...
0
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1answer
51 views

Calculating joint probability problem

This is the example from Casella and Berger 2nd page 146. you have $$f(x,y)=e^{-y}, 0<x<y< \infty $$ and you are interested in finding $$P(X+Y\ge1)$$ They solved this problem the following ...
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0answers
24 views

Integrability of a sequence of iid random variables

I'd really appreciate some hints on the first part of the following question: Let $f_n, n\in \mathbb{N}$ be a sequence of iid random variables over $(\Omega, A,P)$. That is, $P(\{f_1 \in ...
1
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0answers
32 views

Sufficient and necessary condition for the integrability of a random variable

Could anyone give me a hint on the following problem? Thanks! Let $f$ be a random variable over a probability space $(\Omega,A,P)$. Show that $f$ is integrable $\iff $ $\sum\limits_{k=1}^\infty ...
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0answers
31 views

Convolve multivariate gaussian

I wish to find $p(\eta_\star)=\int_{}p(\eta_\star|\eta)p(\eta)d\eta$. Where $p(\eta_\star|\eta)=\mathcal{N}(a^T\eta,\Sigma_1)$ and $p(\eta)=\mathcal{N}(\mu_2,\Sigma_2)$. The problem I have is really ...
2
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1answer
58 views

$E(x^k)$ under truncated $\mathcal{N}(\mu,1)$

There is a similar question in $E(x^k)$ under a Gaussian. However, it doesn't seem to be trivial when $\mu\ne0$. As mentioned in the previous question $k$ is not an integer. The integral that I need ...
0
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0answers
66 views

Expectation under similar distribution to inverse Gaussian

I have the following integral that I wish to evaluate: $\int_0^\infty x\,\, p(x)dx$. Where, $p(x)\propto \exp\left(-ax+\frac{b}{x+k}\right)$ for $a,b\ge0$ Firstly what would the normalising constant ...
0
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1answer
117 views

Integral in Stata when upper limit is infinity [closed]

How can we calculate the integral of the integrand with the lower limit 0 and upper limit infinity in Stata? I am aware of the integ command, but I am not sure ...
4
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0answers
103 views

Integrating a Gaussian Distribution multiplied by 1/x^(2/3), any ideas?

I am trying to integrate the following function over $x$ and $g$: $f(z)= \int_{a/z}^{d} \int_{a}^{zg} \frac{e^{-(g-\mu)^2/2\sigma^2}}{x^{2/3+1}} dx~dg$ When I integrate over $x$ I get: $f(z)= 3 ...
2
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2answers
194 views

Improved Monte-Carlo method vs. hit-and-miss method

I do not understand Which is more accurate, the hit-and-miss method or the improved Monte-Carlo method? Here it is written that that the hit-and-miss has a higher variance but they showed ...
2
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1answer
73 views

Find the pdf of Y

$$ f(x)=\frac{1}{2}e^{-|x|} , -\infty < x < \infty ; Y=|X|^{3} \ $$ I understand that I have to divide it in two parts and write it in cdf form $$ F_{x}(y^\frac{1}{3}) - F_{x}(-y^\frac{1}{3}) ...
2
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0answers
97 views

Integration step in Bayesian linear regression

I want to obtain the integral: $$\int_{{\mathbb R}^p} \frac{1}{(2\pi)^{\frac{n}{2}}\vert\Sigma\vert^{\frac{1}{2}}}\exp\left[-\frac{1}{2}({\bf y} - {\bf X}{\beta})^{\top} \Sigma^{-1}({\bf y} - {\bf ...
1
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4answers
120 views

For what broad types of statistics are integral calculus useful? [closed]

I understand how differential calculus is useful for basic Maximum Likelihood estimation techniques. However, my question is: what broad types of statistics require an understanding of integral ...
2
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1answer
115 views

Sampling from marginal using integrated conditional

I would like to sample from: $$ p(\theta_2|x)=\int p(\theta_2|\theta_1,x) . p(\theta_1|x) . d\theta_1 $$ knowing that I can easily sample from $p(\theta_1|x)$ and (less easily) from ...
3
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2answers
177 views

why isn't the the marginal distribution needed when using a conjugate prior?

What is a good explanation as to why you wouldn't have to integrate to find the posterior when you use a conjugate prior. Most examples (like for instance: http://www.youtube.com/watch?v=0XD6C_MQXXE) ...
1
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0answers
78 views

Can the proof of the central limit theorem be expressed as the limit of a convolution with an operator

The central limit theorem, as I understand it (engineer, non-statistician), says that the distribution comprised of means of some other reasonably behaved distributions converges to a normal ...
1
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0answers
116 views

Finding the integral of a fitted function

I have a function obtained by fitting some data, and I do not have access to the data itself. The fitting parameters of the function have confidence bounds. I need to obtain an expression for the ...
5
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2answers
232 views

Mean of log of cdf

Let $CDF$ be the cumulative distribution function for the standard normal distribution. Let $Z$ be a standard normal random variable. Then $CDF(Z)$ is uniformly distributed on the unit interval, so ...
7
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1answer
156 views

Vector calculus in statistics

I'm teaching a class on integration of functions of several variables and vector calculus this semester. The class is made up most of economics majors and engineering majors, with a smattering of math ...
3
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3answers
806 views

Values for integral of square of standard Brownian process

I am trying to generate values in a table for the following function: $$ W = \int_0^1 [B(t)]^2 dt $$ Where $B(t)$ is a standard Brownian motion. Example: $W_{0.05} = 1.656$, $W_{0.025} = 2.135$. ...
1
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1answer
69 views

Replicating integration results from a paper

I’m reading Wacek’s paper Parameter Uncertainty in Loss Ratio Distributions and its Implications and trying to figure out how to replicate some of the results. Table 6, on page 190 of the paper, ...
1
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1answer
268 views

Confidence Interval in Monte Carlo integration

I want to integrate $\int_{\mathbb{R}_+}\mathbb{1}_A(x) d\mathbb{P}(x)$, in other words I am interested in $\mathbb{P}(A)$. I did this numerically with two Monte Carlo steps. First, I drew, say a ...
1
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1answer
237 views

Integral of a conditional uniform distribution leads to improper integral

I have two uniforms distributions, $X_1 \sim\it{U}(a,b)$ and $X_2\sim\it{U}(X_1+\delta,b+\delta)$. I would like to compute $P(X_2\in[a+\delta,b+\delta])$. So I do this: $$\begin{eqnarray*} ...
4
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1answer
187 views

Is output of Deamer deconvolution not a density?

I have a Model Y= X+e and need the density of X. The deamer package deconvolves the density for X, but if I use the simpsons rule to integrate this density, I get values which are above 1. The ...
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1answer
462 views

Expected log value of noncentral exponential distribution

Suppose $X$ is non-central exponentially distributed with location $k$ and rate $\lambda$. Then, what is $E(\log(X))$. I know that for $k=0$, the answer is $-\log(\lambda) - \gamma$ where $\gamma$ ...
1
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1answer
160 views

How to calculate a posterior for the given model?

Suppose we have a joint distribution on vector $[\mathbf{x}, y]$: $$ p([y, \mathbf{x}] ) = \mathcal{N}\left(\begin{pmatrix} y \\ \mathbf{x}\end{pmatrix}| 0, \begin{pmatrix} k& \mathbf{v} \\ ...
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0answers
92 views

Differencing weekend fluctuations with R?

Suppose a time-serie like this on the left-top corner with weekend and daily fluctuations. This time-series need differencing due to the rising ACF (bottom-left) and portmanteau tests' p -values too ...
3
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1answer
419 views

Monte carlo integration in spherical coordinates

I was playing around with writing a code for Montecarlo integration of a function defined in spherical coordinates. As a first simple rapid test I decided to write a test code to obtain the solid ...
0
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0answers
85 views

Help evaluating a posterior probability expression

Consider $\boldsymbol{x}= [x_1,x_2,...x_n]$ and $\boldsymbol{y}= [y_1,y_2,...y_n]$ to be two multivariate Gaussians with an isotropic diagonal variance structure and uninformative priors so that: ...
10
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1answer
489 views

What is the expected value of modified Dirichlet distribution? (integration problem)

It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If: $ X_i \sim \text{Gamma}(\alpha_i, \beta) $ Then: $ ...