The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
0answers
17 views

Uncertainty on ratio of integrals (correlated)?

I have a histogram, and a quantity defined as the integral of a subrange of that histogram (int_sub) divided by the integral of the full range (...
0
votes
0answers
23 views

What regression to use for integral and scale data?

I am hoping to compare reaction times with ratings from a questionnaire. The ratings on the questionnaire could range from -9 to +9 although in the data as expected they are only positive results ...
0
votes
1answer
31 views

Given a sample of size n from a normal distribution, estimate the probability of picking a value larger than X from this distribution

Say I picked 10 random samples from a normal distribution with unknown parameters: 1.7, 2.6, 3.0, 4.4, 1.6, 2.1, 2.4, 2.7, 5.2, 3.3 What is the probability that I will pick a value larger than ...
4
votes
1answer
74 views

Gibbs Sampler transition kernel

Let $\pi$ be the target distribution on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R^d}))$ which is absolutely continuously wrt to the $d$-dimensional Lebesgue measure, i.e : $\pi$ admits a density ...
0
votes
0answers
16 views

Conditional Probability in Multivariate Normal

Given a tri-variate Normal, the conditional probability of an element given others truncated information is Now if I know that the mean vector u is (-0.91,-1.31,-1.39) and R is ...
0
votes
1answer
27 views

indicator variable - dirac delta or step function

I am trying to solve the following equation, \begin{equation} = \int_{-\infty}^{\infty} \frac{1}{\sqrt{ (2\pi)^{k_{Y}} | \Sigma |}} \cdot \mathrm{exp} \{ -\frac{1}{2} (Y - Xm)^{T} \Sigma^{-1} (Y ...
12
votes
3answers
1k views

“The total area underneath a probability density function is 1” - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I ...
5
votes
1answer
138 views

Conditional independence iff joint factorizes

I have proven that: $X⊥Y|Z\ {\rm iff}\ p(x,y|z)=p(x|z)p(y|z)$ for all $x,y,z$ such that $p(z)>0$. The next question is to prove an alternative definition: $X⊥Y|Z$ iff there exist functions $g$ ...
5
votes
1answer
81 views

Truncated Von Mises-Fisher distribution

I am putting a von Mises-Fisher prior on my data. The data does lie on a unit sphere, but the only problem is that my data is always positive. So I feel like I am wasting my prior on unnecessary ...
1
vote
0answers
28 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
votes
0answers
87 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
votes
2answers
56 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
votes
2answers
73 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
votes
1answer
30 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
vote
1answer
64 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
vote
1answer
51 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
votes
2answers
142 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
vote
0answers
119 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
votes
1answer
61 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
votes
1answer
202 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
vote
3answers
159 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
vote
0answers
92 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
vote
0answers
118 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
votes
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
votes
1answer
73 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
vote
2answers
82 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
vote
1answer
64 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
votes
1answer
95 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
vote
2answers
52 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
votes
1answer
56 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 ...
1
vote
0answers
25 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
vote
0answers
35 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 ...
2
votes
1answer
61 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
votes
1answer
180 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
votes
0answers
127 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
votes
2answers
302 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
votes
1answer
76 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}) ...
3
votes
0answers
139 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
vote
4answers
130 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
votes
1answer
119 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
votes
2answers
217 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
vote
0answers
81 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
vote
0answers
129 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
votes
2answers
261 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
votes
1answer
184 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
votes
3answers
1k 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
vote
1answer
70 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
vote
0answers
327 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
vote
1answer
255 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
votes
1answer
214 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 ...