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0
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0answers
6 views

Estimation of parameters from a complicated expression (double integration) with optime [migrated]

I am trying to estimate paramters of a function with optime This is the code: ...
4
votes
2answers
85 views

Integration of product of functions of “x” with exponents/powers (binomial problem)

Original problem: A point X is randomly chosen from the interval (0,1). Suppose X=x is observed. Then a coin with P(Heads) = x is tossed independently n times. Let Y be the number of heads in n ...
3
votes
1answer
53 views

Expectation of two identical lognormal distributions

I would like to compute the conditional expectation (on an interval from $c$ to $\infty$) of the minimum of two log normal distributions. Denote $X_1$, $X_2 \sim LN(0, \sigma)$, the associated ...
1
vote
1answer
43 views

How to show $\int_{-\infty}^{\infty}f_{X}\left(x\right)\left(P\left(X<x\right)-P\left(X>x\right)\right)dx =0$

For a continuous random variable X , intuition tells me that $$\int_{-\infty}^{\infty}f_{X}\left(x\right)P\left(X<x\right)dx=\frac{1}{2}$$ and more weakly that $$\int_{-\infty}^{\infty}f_{X}\left(...
2
votes
1answer
36 views

Help calculating integral

I need help calculating a integral. It is a Pareto distribution with common tail $\frac{1}{(1+x)^a}$, where I assume countermononicity between two variables $X_1$ and $X_2$. $\int_0^1 ( (\frac{1}{1-x}...
4
votes
1answer
44 views

Question about accuracy in Monte Carlo integration

Suppose that we want to estimate the integral: $$\psi=\int_{a}^{b}h(x)dx.$$ Let $\hat{\psi}$ be the Monte Carlo estimator. As far as I know, if we desire an accuracy up to the fourth decimal, we need ...
2
votes
1answer
27 views

Marginal Posterior distribution with Normal observations

According to chapter 3 of Gelman's Data Bayesian Analysis[DBA], when we have $y_i\sim N(\mu,\sigma^2)$, and $p(\mu,\sigma^2)\propto (\sigma^2)^{-1}$ Then, $p(\mu,\sigma^2|\mathbf{y})\propto \sigma^{-...
2
votes
1answer
22 views

Proving the validity of a kernel

How can we prove that the following is a valid kernel? Let $\phi$ be any function on R X R. Define: $K(x,y) = \int{\phi(x,z)\phi(y,z)dz}$ We want to show that $K$ is a valid kernel.
3
votes
2answers
71 views

Expectation of log likelihood ratio

Given that $X_{1},...,X_{n}$ are i.i.d random variables with joint distribution $f(x\mid \theta) $ with 1 dimensional parameter $\theta$, let $\hat\theta$ be the maximum likelihood estimator of $\...
7
votes
1answer
121 views

Example of computing the expectation of a discrete RV using Riemann-Stieltjes integral?

Riemann-Stieltjes integral notation is used in expectation expressions in some probability texts. Basically, dF(x) pops up in the integral rather than f(x)dx in the integral, since the CDF F(x) may ...
0
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2answers
39 views

How to express p(y|x) in terms of p(x,y)?

Given two inter-dependent random variables $X$, $Y$ with a joint probability density p(x,y), can we express the conditional probability $p(y|x)$ in terms of the integral of something related to $p(x,...
0
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0answers
36 views

Does the LHS of $E[X_n | \mathscr F_{n-1}]$ make sense even if $X_n$ is not integrable or adapted?

Let $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ be a filtered probability space. Then $X_n$ is a $(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)-$martingale if: $X_n$'s ...
0
votes
0answers
29 views

Is the following integral of a pdf an identity, i.e. always true?

I am reading a paper and the author starts a proof with this $$ p(\hat{R}|R) = \int p(\hat{R},\theta|R)d\theta $$ p is the density function. Is this something that is always true? Can you help me ...
2
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0answers
101 views

Prove $\int_E |f| d\mu < \infty$, $\lim \int_E f_n d\mu \to \int_E f d\mu$

Given a measure space $(\Omega, \Sigma, \mu) = (\mathbb R, \mathscr B(\mathbb R), \lambda)$ (Or: a similar probability space on some subset of $\mathbb R$ that has Lebesgue measure 1). Let $E \in \...
2
votes
1answer
120 views

How can I calculate this integral? $\int_{-\infty}^{\infty} \Phi (a + bX) \phi (c + eX) dx$

Suppose we have the density and distribution of the standard normal. How can one calculate the integral: $\int_{-\infty}^{\infty} \Phi (a + bX) \phi (c + eX) dx$ Note this is not included in the ...
2
votes
2answers
100 views

Prove $E[|Z|] < \infty \to E[|Z_n|] < \infty$

Let $Z$ be an integrable random variable on filtered probability space $(\Omega , \mathscr F, (\mathscr {F_n})_{\{n \in \mathbb{N}\}}, \mathbb P)$ Define $Z_{n} := E[Z|\mathscr {F_n}]$. Show that $...
4
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0answers
43 views

how does one compute expectations for non-linear functions

I am continuing my struggles with approximate Bayesian inference methods. I have a fundamental doubt about how to compute certain expectations that arise during variational bayes, for example. So, my ...
5
votes
1answer
111 views

Multivariate probability density - Help with an integral

I'd like some help solving this problem about multivariate probability densities. Let the random variables X and Y have the joint density f(x,y) = 1/y for 0 < x < y < 1 and 0 otherwise....
2
votes
1answer
47 views

Finding the characteristic function of $Y \sim U(-1,1)$

I know that $\phi_Y(t) = E(e^{itY})=E(\cos(tY))+iE(\sin(tY))$ After integration I have found that $E(\cos(tY))= \frac{\sin(t)}{t}$ and $E(\sin(tY))=0$. So is the characteristic function just $\frac{\...
0
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0answers
11 views

Why does the moment generating function of a chi-squared random variable only exist for t<1/2?

I have found that for a chi-squared ($n$ degrees of freedom) random variable $X$, $M_X(t)= (1-2t)^{-n/2}$. I am told that this only exists for $t<1/2$. Why is this?
1
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0answers
42 views

Bayesian inference on gamma distribution

The likelihood of an observation $x$ under a gamma distribution is $$L(x | \alpha, \beta) \propto \beta^\alpha x^{\alpha-1} \frac {\exp(-x\beta)} {\Gamma(\alpha)}$$ Suppose I have some observations ...
0
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0answers
27 views

Integral involving normal CDF

Let $\Phi$ be the normal CDF. I have calculated and plotted numerically the following integral $$f(c)=\int_{-\infty}^0 \vert (1+c)\Phi(x/(1+c))-\Phi(x)\vert dx,$$ for $c\in(0,1)$. However, I would ...
-1
votes
1answer
34 views

Evaluating an integral

Just want to make sure of something very simple. $X$ follows a distribution $F(\mu,\sigma^2)$. I want to calculate the integral of $XdF(\mu,\sigma^2)$ where $\mu=E(X)$ and $\sigma^2=Var(X)$. Do I ...
0
votes
0answers
34 views

Is there a monte carlo guarantee without the normalization?

There are all kind of monte carlo integration techniques such that an expected value $E[f(X)]$ is estimated by using $\frac{1}{n} \sum_{i=1}^n f(x_i)$ for $x_i$ being samples from the underlying ...
3
votes
1answer
87 views

determine constant for biweight (bisquare) function

In robust statistics a biweight (bisquare) function is defined as follows $$\rho \left( x \right) = \gamma\left( {1 - {{\left( {1 - {{\left( {\frac{x}{c}} \right)}^2}} \right)}^3}} \right){{\bf{1}}_{...
2
votes
1answer
250 views

Overlap between two normal pdfs [duplicate]

I have two normally distributed random variables (estimated from two different sets of samples), and I'd like to know how "similar" those variables are (in order to compare the sets). I had the idea ...
0
votes
2answers
155 views

How to deal with 'cut-off' selection bias/sampling bias? (truncated distribution)

In short When measuring an outcome with a normal distribution, but whos mean is below the detection threshold, can you still make statements about differences between populations? Example Say I ...
1
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0answers
83 views

How to calculate E(1/Y) when Y is Inverse Gaussian distributed?

The Inverse Gaussian Distribution density is : $$\frac{\phi^{\frac{1}{2}}}{\sqrt{2\pi y^3}} exp[\frac{-\phi(y - \mu)^2}{2\mu^2y}]$$ Got to this integral: $$\int_0^\infty \frac{1}{y} \frac{\phi^{\frac{...
3
votes
1answer
65 views

Distribution of stochastic integral

I would like to find the distributions of the following random variables: $Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process. What is the ...
0
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0answers
26 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
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0answers
27 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
71 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 100....
5
votes
1answer
146 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 $\pi(x_1,....
0
votes
0answers
30 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
67 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 ...
17
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3answers
4k 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
291 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 $g$ ...
7
votes
1answer
161 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
31 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
97 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
189 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
75 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
36 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
76 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} p(S_{t+...
1
vote
1answer
76 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
212 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)= \displaystyle\...
1
vote
0answers
293 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
76 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
563 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
227 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.