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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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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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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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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49 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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How to find the pdf from time series

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
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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
105 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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Plotting of density estimates in matlab [migrated]

Given the biometric match scores, I am required to plot the graphs of estimated densities of matching genuine and impostor scores. Following are the graphs I got for genuine and impostor scores ...
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1answer
40 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
155 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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39 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
47 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 ...
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1answer
42 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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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
47 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 ...
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1answer
107 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
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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
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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 ...
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36 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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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
26 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
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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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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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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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35 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
30 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
230 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
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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 ...
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1answer
73 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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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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540 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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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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PDF for ratio of Dirichlet

I have a random vector $\theta$, which was generated by: X1 ~ Dirichlet($\alpha_1$) X2 ~ Dirichlet($\alpha_2$) $\theta$ = ...
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1answer
111 views

Sum of normal truncated random variables

Suppose I have $n$ independent normal random variables $$X_1 \sim \mathrm{N}(\mu_1, \sigma_1^2)\\X_2 \sim \mathrm{N}(\mu_2, \sigma_2^2)\\\vdots\\X_n \sim \mathrm{N}(\mu_n, \sigma_n^2)$$ and ...
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1answer
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Scale parameter MLE scheme known but how to find according distribution PDF?

For known location, we can find the scale parameter of a normal distribution by calculating the sum of squared differences to the location, then dividing by n-1 and taking the square root. This is the ...
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How to calculate CDF of g(X)

Let $X$ a random variable with distribution $F_X(x)$ $$Y=g(X) = \left\{ \begin{array}{lr} X-c & : X > c\\ 0 & : -c < X \le c \\ X+c & : X \le -c \end{array} \right\}$$ ...
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1answer
50 views

Rectifier function of random variable

Let $X$ be a random variable with distribution $F_X$ and density $f_X$. Define $$g(x) = \left\{ \begin{array}{lr} x & : x \ge 0\\ 0 & : x < 0 \end{array} \right\}$$ and let ...
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1answer
57 views

Identifying subsets for outlier detection in local outlier factor

I am trying to gain better understanding of the idea of local outliers (as discussed in this pdf) and how the function is implemented. Here are the key passages from the pdf: Local outliers: ...
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1answer
83 views

Non-central scaled Student's t cumulative density function required (alternatively the pdf)

I need to cite the pdf(density) or cdf(distribution function) of a non-central scaled Student's t distribution. There is an article about the non-central Student's t distribution ...
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1answer
71 views

Probability function of three Random Variables multiplied, solidifying intuition

I ran across this exercise: Let $T$ be a random variable distributed as a $\text{Bernoulli}(p)$, $U$ be a random variable distributed as a $\text{Bernoulli}(q)$ and $W$ be a random variable ...
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1answer
111 views

Why likelihood is not always a density function? [duplicate]

I try to self-learn Bayesian machine learning (mostly by studying Bishop and Kevin Murphy's books). While working with formulas I was puzzled by the quote that "Note that the likelihood function is ...
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38 views

Moment generating function of a distribution

I want to find the moment generating function (mfg) and mean deviation of this distribution: $$f(x,\epsilon,k,\theta) = k\theta^{(1+1/k+\epsilon/k)}x^{(k+\epsilon)}\exp{(-\theta x^k ...
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1answer
124 views

Computing inverse probability weights — conditional (multivariate) density estimation?

The general version: I need to estimate $f(A | X)$ where $A$ and $X$ are continuous and multivariate. I'd rather do it nonparametrically because I don't have a good functional form in mind and ...
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Normal Probability Density Function and confusion over how it arrives at an answer [duplicate]

I am confused at how the normal distribution's PDF capable of calculating a density for a single variable. I understand that the CDF probability of an exact continuous random variable $X$ is 0. ...
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1answer
47 views

How do I compute the density of this distribution

Ferdosi et al define six artificial datasets to test density estimation methods. Part of the fourth dataset is defined as: $$ Uniform(x,y) = [0,100], Gaussian(z) = [M = 50, var = 5] $$ Where $M$ is ...
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Goodness of fit of an approximation of a PDF

I'd like to evaluate the performance of an algorithm which learns to approximate a discrete PDF for a latent variable given some noisy input. Now, is there a standard test to evaluate the goodness of ...