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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How does one analytically solve for the contours of an arbitrary bivariate density?

This slide deck shows how to solve for the contours of a multivariate normal, using an eigendecomposition of the covariance matrix. This seems to rely on the convenient ellipsoid shape of the normal's ...
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Density plot of parameter estimates from linear regression model

I am running a linear regression model in R: data(iris) fit1.iris = lm(Sepal.Length ~ Petal.Length+Petal.Width , data=iris) summary(fit1.iris) These are my ...
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34 views

Proof of probability density function? [closed]

Is there any website or any book which you can recommend to me for learning the proof of all PDF (distribution function)?
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72 views

Mean and variance of ranks

Consider rank data 1 to n with two groups, n=n1+n2, how would one test the null that the two groups have equal rank distributions using MOMENTS? (Wilcoxon is not the answer) Is MLE possible to do the ...
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153 views

Assume $X\sim N(\mu,\sigma^2)$. What is the pdf of $X^3$

I have the following that is remaining unanswered and would love some help: Assume $X\sim N(\mu,\sigma^2)$. What is the pdf of $X^3$. For a large sample, $n$, what is the variance of the cube of the ...
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Showing a joint function is a pdf and joint MGF

I have these two problems that are on a previous final for my class. for Number 3, I know that the double integral from 0 to infinity of f(x+y)/x+y dxdy has to equal 1. But I have no idea how to ...
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1answer
26 views

Multiplication of two random distribution

I am trying to find the resulting PDF , when two random functions are multiplied. First function obeys normal distribution and second function obeys cauchy distribution. Can anybody tell me how to ...
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Guidelines to estimate using MLE from this definition of error function

Consider a stable causal, single-input/single output, linear time-invariant, discrete-time system. The noisy output is $y[n] = \sum_{i=0}^{p-1} c_i d[n-i] + w[n]$ where $c_i$ is the real-valued ...
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24 views

Help in finding the pdf for Gaussian distribtuion of time series model

PROBLEM STATEMENT: The original data $y_t$ is a noisy version of a time series obtained from an autoregressive process excited by a deterministic non-linear signal $x_t$. The error terms $u_t$ is : ...
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37 views

Density function for non-gaussian case

Consider a linear time series model - Autoregressive model ($y$) of dimension $n \times 1$ where $n$ is the number of data samples and only one variable AR model of order $p >1$ is excited by ...
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42 views

What is the probability density function of this arbitrary signal

Lecture slide SLide#10, mentions how to generate Pseudo Random binary sequence (PRBS). I says that we can take the sign of a white, zero-mean Gaussian noise signal to form the PRBS. What will be the ...
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Maximum likelihood estimation involving both probabilities and probability densities

Note: based on suggestions in the comments, I have rewritten this question. Please refer to the history for the original version. In general my question regards how to compute likelihoods in mixed ...
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35 views

Finding Expectation of a Random Variable Using Its Joint Marginal Density

If X and Y have joint density function $$ f(x,y) = \frac{1}{y} \,\mathbb{I}_{0<x<y<1}, $$ how do I find the expectation of X or Y? Since E[X] requires us to know the PDF of X, I tried to ...
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32 views

PDF of Distance between the centre of a regular hexagon of radius R and any point within it

What is the probability distribution function of the distnace between the centre of a regular hexagon of radius R and any point within it? I have done the following and would appreciate if you could ...
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Are there non-trivial settings where the MAD statistic has a closed-form density?

The MAD statistic of an iid sample $(x_1,\ldots,x_n)$ is defined as the median of the absolute deviation from the median: $$ ...
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Unable to understand joint pdf and EM

I am unable to understand how the density function is derived in this paper Semiblind System Identification with Symbolic Chaotic Sequences The Authors have ...
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115 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 ...
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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 ...
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18 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 ...
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55 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 = ...
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125 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 ...
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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. ...
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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 ...
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32 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 ...
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60 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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47 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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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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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 ...
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84 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 ...
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1answer
51 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 ...
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40 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: ...
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72 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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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 ...
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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
24 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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28 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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18 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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38 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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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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87 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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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 ...
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76 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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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 ...
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1answer
68 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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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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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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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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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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32 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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24 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 ...