Tagged Questions

PDF stands for Probability Density Function. The PDF of a variable gives the relative probability for each value of a continuous variable. Use this tag when asking about probability functions in general, whether PDFs or discrete probability mass functions.

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3answers
106 views

Fast density estimation

Suppose you are trying to estimate the pdf of a random variable $X$, for which there are tons of i.i.d. samples $\{X_i\}_{i=1}^{n}$ (i.e. $n$ is very large, think thousands - millions). One option is ...
0
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0answers
16 views

Calculate pdf of complex model

I'm trying to model the distribution of effects of mutations (let's call it s) in evolution but I'm stuck in generating the probability distribution function (pdf) for my model. So, my model is a ...
1
vote
0answers
19 views

Density function for ARMA

There is a model expressed as $y_t = h^Ty_{t-1} + b^T x_t$ where $x$ is a zero mean uncorrelated white Gaussian noise of unity variance $\sigma^2_x$. This appears to be an ARMA model. But, I am ...
0
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0answers
14 views

Create a density map with latitude and longitude

I am working on a data mining project for my class and would like to visualize some data. I basically have 20,000 instances and would like to create a density map based on house value. I have ...
1
vote
1answer
39 views

Unable to calculate the density function for AR

The model is an AR(p) process excited by a white Gaussian noise $\epsilon_t$, \begin{align} Y_t = &c+ \phi_1Y_{t-1} + \phi_2 Y_{t-2}+ \ldots+ \phi_p Y_{t-p} + \epsilon_t\\ \epsilon_t = ...
4
votes
1answer
111 views

Why is my R density plot a bell curve when all datapoints are 0?

When I graph a density plot in R, and all the numbers are slightly greater than 0, I get essentially a vertical line at x = 0. But when all the numbers are exactly ...
1
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0answers
15 views

Probability denisty around a linear regression line

In a very basic problem or linear regression y= mx + q one can define an Confidence Interval around the line. This has a characteristic shape: narrow at the center getting bigger at the extremities. ...
1
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0answers
20 views

A confusion on a continuous probability problem

We are given a continuous Joint pdf $$f(x,y)=\frac23(x+1), 0<x<1,0<y<1$$ We are to find $\textsf{P}(X<2Y<3X)$ What I did : It is easy to see that the pdf's are independent and ...
0
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0answers
22 views

get probability density from characteristic function (inversion theorem)

I am currently trying to use the inversion-theorem to get the density from a characteristic function. For people familiar with finance, I am trying to get the density of a stochastic volatility model ...
0
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0answers
12 views

How to find the pdf in higher dimension

I have a time series output $Z = X - Y$ where the dimension $d$ of $Z$ is 1D. I don't know what process generated the signals $X,Y$. All I have information is that $X$ is corrupted with additive ...
4
votes
1answer
52 views

How can I evaluate the probability density function of $Z=X+Y$,if $X$ and $Y$ are not independent?

If $Z=X+Y$, and the PDFs of $X$ and $Y$ are both functions of a deterministic variable $d$, how can I evaluate the PDF of $z$ while the convolution cannot be used here (due to lack of independence)?
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1answer
42 views

How to sample discontinuous probability functions?

I fitted a pdf with CumFreq to data, and the best fitting pdf is a discontinuous function composed as: ...
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0answers
35 views

What is the mean and variance of this probability distribution?

I don't know if this distribution has a name or not so the best I can do is describe how it is obtained. You start with two arrays of n uniform random variables U(0,1) where n>2. I will provide a ...
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0answers
12 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
4
votes
2answers
77 views

Does the central limit theorem apply to these probability density functions?

Let's say you have n uniform random variables from 0 to 1. The distribution of the average of these variables approaches normal with increasing n according to the central limit theorem. What if ...
2
votes
1answer
47 views

More flexible bell shape than log normal distribution

I am looking for a very flexible bell shape function, with asymmetry on both sides of the bell, also with the possibility that the left arm of the bell had a milder slope while the right had a steep ...
0
votes
0answers
26 views

Kernel density estimation on bounded support

I was looking for some way to deal with boundary bias of kde in case of unit interval. One example is an usage of Chen estimators (or Beta estimators; an example might be seen here: ...
5
votes
1answer
67 views

Assuming a probability density for MLE to do model selection

Motivation: I am trying to use Akaike Information Criterion to assess model ranking and over-fitting risk for a set of nonlinear models. I am an electrical engineer with no formal statistical training ...
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votes
0answers
12 views

The validity of using truncated PDFs as prior distributions?

I am trying to implement an ABC (Approximate Bayesian Computation) rejection-sampling algorithm in R. I am currently working with a six-parameter model and for each of the parameters I have specified ...
0
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1answer
23 views

Joint PDF of a set of equations

I am looking for a way to find the joint pdf of vector $Z=[Z_1,Z_2,Z_3,Z_4]$ where $Z_1= a_1 X_1^2 + a_2X_1Y_1+ a_3 X_1Y_2 + a_4Y_1^2 + a_5Y_2^2$ $Z_2= b_1 X_1^2 + b_2X_1Y_1+ b_3 X_1Y_2 + b_4Y_1^2 + ...
0
votes
1answer
22 views

Sum of dependent R.V

I have two random variables whose PDF are parameterized by an unknown constant as follows: P(A;d) P(B;d) apparently, these two are not independent, so to find P(A+B;d) one cannot use convolution. ...
1
vote
1answer
25 views

PDF of sum of independent Gaussian variables

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
0
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0answers
17 views

Show about an arbitrary measurable function that has density

This is an exercise that I have to finish and I hope you to be patient maybe I violated the rules since that it is not always possible to start solution of an exercise. Maybe can be very useful only ...
1
vote
1answer
37 views

Stochastic Volatility Model

In Kim et al. (1998) stochastic volatility model is specified as: $y_t=\beta\exp({\frac{h_t}{2}})\varepsilon_t,\quad t\geqslant1$ $h_{t+1}=\mu+\phi(h_t-\mu)+\sigma_\eta\eta_t$ $h_1\sim ...
2
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0answers
21 views

How to simulate a sample with density $f^p / \int f^p$?

Given a density $f$ (or given a $n$-sample $X_1, \ldots, X_n$ with density $f$), is there a way to create a $n'$-sample $X_{p,1}, \ldots, X_{p,n'}$ with density $f^p / \int f^p$, where $p \in ...
1
vote
1answer
63 views

Elegant way to plot a probability density function?

Given a set of samples I would like to draw a nice plot showing their probability. Something like this (notice the vertical bar showing the samples): Or even harder like this (considering a weight ...
1
vote
1answer
16 views

Obtaining a sample with an given sample (resulting) covariance matrix

Often, we are interested in generating data from a density $ f(x \vert \boldsymbol\theta) $, with data $x$ given some parameter vector $\boldsymbol\theta$. This results in a sample, from which we may ...
2
votes
1answer
63 views

Convolution of random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
0
votes
0answers
17 views

Want to findout satistical distance unsing another procedure except kullback liebler divergence

i want to find the distance between two pdf(pdf is calculated using kernel density estimator from two random data set of different size ) .Is there any alternative and efficent way to calculate ...
2
votes
1answer
67 views

What would be the likelihood function of a pdf, $p(n)=1-|n|$ for $|n|<1$?

This might seem like a basic question to some but I am utterly confused by the fact that the given pdfs are not Gaussian or any other distribution commonly seen in examples. I have two hypotheses ...
1
vote
1answer
36 views

The sum of the kernel density values is not 1?

>> x = [randn(30,1); 5+randn(30,1)]; >> [f,xi] = ksdensity(x); >> sum(f) ans = 5.5376 I ran the ...
2
votes
0answers
29 views

Change of variables with a continuous but not differentiable mapping

Suppose that Y is a continuous random variable with a density function $f_{Y}(y)$. We transform $Y$ by the following mapping \begin{equation} Y^{*} = \left \{ \begin{array}{ll} \alpha Y + \beta ...
0
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0answers
32 views

Absolutely continuous probability distribution and its probability density

A Wikipedia article states: A random variable $X$ has density $f_X$, where $f_X$ is a non-negative Lebesgue-integrable function... $F_X$ is the cumulative distribution function of $X$... ...
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0answers
23 views

Estimating time-lagged mutual information for two signal samples

This is an attempt to reproduce Moon et al. 1995, and the author's copy can be obtained through here. As a benchmark, we estimate the time-lagged mutual information of a simple sine signal ...
0
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0answers
30 views

Approximate Probability Distribution Function

I am trying to approximate a large discreet probability distribution function using a histogram with a small number of entries. I.e., create a piece-wise first-order polynomial approximation for a ...
0
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0answers
20 views

Spatially explicit capture–recapture with R: density estimation of transect detectors in SECR

Dear stakexchange community I am new here and I hope to get an answer about density estimation of transect detectors in SECR. To my data: I collected samples of individuals at eight transects. Data ...
1
vote
1answer
74 views

Probability distribution estimation — why normalize by bin width?

This is from a typical introduction to kernel density estimation. Suppose we want to estimate the probability density function $p(x)$ given a set of samples $x_1,x_2 \ldots x_N$. The simplest method ...
3
votes
0answers
24 views

Error bounds when approximating densities

I was curious whether it is possible to obtain approximation error bounds on continuous densities from approximation error bounds of random variables. To make my question more precise: We consider ...
2
votes
1answer
113 views

maximum-likelihood of a sequence of events described by a Bernoulli distribution

I am having quite some troubles with the following homework: In a city it's measured for the whole year whether it rained or not. A distribution $\textrm{Bernoulli}(r_t|\rho)$ characterizes the ...
2
votes
1answer
46 views

Sum of Random Variables

As part of my statistical mechanics class, I'm trying to go through Kardar's statistical physics of particles and I'm having trouble with this one line: Consider the sum $X=\displaystyle ...
4
votes
1answer
196 views

Do mean, variance and median exist for a continuous random variable with continuous PDF over the real axis and a well defined CDF?

For a continuous random variable with continuous PDF over the real axis and well defined CDF, are the mean, variance, and median always well defined? Mean and variance do not always exist, e.g. for a ...
0
votes
0answers
40 views

E-mail answered probability after n days of waiting for a reply - based on a sample of e-mails and replies

Here is the task: I have a sample of replies to my e-mail from my mail box. A sample is taken over a period of 90 days, 1000 e-mails and replies if any. (We only consider a pair of {my original ...
2
votes
1answer
49 views

Value for which the PDF(value) is maximal in a distribution with skew?

I am working on a project where I need to chart statistical data and related, skewed distributions a la http://en.wikipedia.org/wiki/Skew_normal_distribution. Unlike with normal distributions, in ...
2
votes
1answer
46 views

How to test if some data points is drawn from a distribution with linear PDF?

I have some data in the range [0, 1], and from the histogram below, it seems that they might be drawn from a distribution with linear probability density function (what's the name of that kinds of ...
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0answers
16 views

Comparing densities of a feature for different classes when the feature is irrelevant to one class

Let us suppose that I have a number of features. I design pdfs for every feature and every class, some of them by smoothing some histogram of training samples, others just by introducing the prior ...
3
votes
2answers
69 views

Gaussian Mixture Model parameters from density

How do I estimate parameters of subpopulations in a 1D gaussian mixture model when I already have density (measured on a grid) of the mixture? All the algorithms I can find (like the well-known EM ...
5
votes
1answer
150 views

How to statistically compare groups for multiple density plots?

Is there a statistical method to compare these density plots other than ANOVA (MANOVA)? I would like to compare the densities among year within each plot and report which of those distributions are ...
1
vote
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
vote
2answers
98 views

How to find pdf of a joint distribution in R?

$F(x,y) =\frac{1}{6}(x^2\, y+x\, y^2)\,,\quad 0\leq x\leq 2,\, 0\leq y\leq 1$ Above is the joint distribution given, how to find out cumulative distribution function of y? how to obtain joint ...
0
votes
2answers
63 views

What does drawing sample using Metropolis-Hastings algorithm mean?

I am confused with the word "draw samples from any probability distribution P(x)", mean I apologize for my ignorance, but, drawing sample as i understand, is for example, tossing a coin and writing ...