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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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 + ...
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1answer
18 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. ...
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1answer
20 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 ...
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16 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 ...
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1answer
36 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 ...
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20 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 ...
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78 views
+50

Help in maximum likelihood estimation derivation

The problem is estimating a metric $Vol(D)$ for the following situation : Given noisy observations the observed data are a random sample $Y_1,\ldots,Y_n$ where $Y_i \in \mathcal{R}^d$. The model for ...
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1answer
52 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 ...
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1answer
15 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
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1answer
52 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 $\ast$ ...
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11 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
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1answer
63 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 ...
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1answer
27 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 ...
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0answers
25 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 ...
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29 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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8 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 ...
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28 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 ...
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12 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 ...
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1answer
60 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 ...
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23 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 ...
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1answer
112 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 ...
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1answer
45 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 ...
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1answer
170 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 ...
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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 ...
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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
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1answer
43 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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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 ...
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1answer
50 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
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1answer
123 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 ...
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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} ...
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2answers
63 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 ...
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2answers
51 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 ...
3
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1answer
39 views

From joint cdf to joint pdf

We can get the joint pdf by differentiating the joint cdf, $\Pr(X\le x, Y\le y)$ with respect to x and y. However, sometimes it's easier to find $\Pr(X\ge x, Y\ge y)$. Notice that taking the ...
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1answer
51 views

How do I compute the density of this data set that is made up of two different 3D-distributions?

A sequel to this question. I have a dataset where: $\frac{4}{5}$ of the points are drawn from: $(x, y) \sim \mathcal{U}_{2}(0,30)$, $(z) \sim \mathcal{U}_{1}(14.5, 15.5)$. $\frac{1}{5}$ of the ...
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1answer
32 views

Numerical approximation of percentiles from arbitrary pdf

Given an easily-computable probability density function $f(x)$, what algorithm can we use to numerically approximate percentiles? For instance, we might be looking for $x$ such that given $X \sim ...
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1answer
18 views

Area under a truncated distribution = 1

I have computed a truncated normal distribution, which total probability density (i.e. area under the curve) is equal to 0.92. The distribution represents well the reality of the phenomenon I am ...
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1answer
44 views

Logarithmic probabilty distribtion and respective problems in matlab

Dear crossvalidated community, quite frankly, I'm not that much into statistics and therefore, I'm having much trouble solving some issues im having in Matlab right now. My plan is to ...
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1answer
30 views

Area below probabilities

Let $p$ be probabilities and $D$ is the real How can I proof that the areas $$\int p \; d F_{p}(p|D=1) = \int (1-p) \; d F_{1-p}(1-p|D=1)$$ are equal. Where $F_{p}$ is the empirical distribution ...
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0answers
41 views

Formalizing pdf using both discrete and continuous densities

I'm trying to formalize the probability density function for a rather simple process, but I'm having difficulty writing it precisely. Specifically, consider simulating a 1-D Gaussian random walk ...
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107 views

How can I calculate a discrete Cumulative Distribution MultiDimensional Array from a discrete Probability Mass Array when dimensions > 2?

I would appreciate any help in trying to calculate the Cumulative Distribution Array of a Probability Mass Array when dimensions > 2, essentially a discrete joint cumulative distribution from a sample ...
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1answer
39 views

Negative density for non-negative variables [closed]

Having an integer positive variable (number of days) in an experiment, I got negative values for the density plots using R. I have read other posts relating to this topic. They admitted that the ...
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1answer
33 views

PDF not matching histogram of synthetic ratios of independent beta

The PDF of the ratios of independent beta variables is described in http://www.tandfonline.com/doi/abs/10.1080/03610920008832632#.U9J02vldUcC To explore the implications, i created an implementation ...
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2answers
235 views

Is there a Bayesian approach to density estimation

I am interested to estimate the density of a continuous random variable $X$. One way of doing this that I learnt is the use of Kernel Density Estimation. But now I am interested in a Bayesian ...
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2answers
74 views

Non-informative prior for regression model

I'm looking at p. 355 of Gelman's Bayesian Data Analysis (3rd ed.), for which there is no errata, and I see this: In the normal regression model, a convenient non-informative prior distribution ...
2
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1answer
100 views

Discrete analog of CDF: “cumulative mass function”?

We call the integral of a probability density function (PDF) a cumulative distribution function (CDF). But what's the cumulative sum of a probability mass function (PMF) called? I've never heard the ...
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3answers
543 views

Do the pdf and the pmf and the cdf contain the same information?

Do the pdf and the pmf and the cdf contain the same information? For me the pdf gives the whole probability to a certain point(basically the area under the probability). The pmf give the probability ...
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4answers
622 views

Calculating PDF given CDF

I know that the PDF is the first derivative of the CDF for a continuous random variable, and the difference for a discrete random variable. However, I would like to know why this is, why are there ...
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1answer
21 views

Laplace transform and density

It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density? ($L_X(z) = E(exp(-Xz))$)
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15 views

Let $f(x)$ be the density of $X$. What is $\lim_{n\rightarrow \infty}\mathrm{E}_X[nf(X+n)]$?

The expectation can be written as $\mathrm{E}_X[nf(X+n)] = \int_{-\infty}^\infty nf(x+n)f(x)\,\mathrm{d}x$. The expectation relies on the speed of tail $f(x+n)$ goes to $0$. I posted a related ...
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30 views

PDF for ratio of Dirichlet

I have a random vector $\theta$, which was generated by: X1 ~ Dirichlet($\alpha_1$) X2 ~ Dirichlet($\alpha_2$) $\theta$ = ...