Questions tagged [pdf]

Probability density function (PDF) of a continuous random variable gives the relative probability for each of its possible values. Use this tag for discrete probability mass functions (PMFs) too.

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5
votes
2answers
4k views

Bivariate Gamma distribution PDF

I'm analyzing a set of data, and I like to fit a gamma distribution. I know how to do it in one dimension, but the data that I'm analyzing now are two dimensional. Is there any way that I can have a ...
10
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3answers
759 views

How to formally test for a “break” in a normal (or other) distribution

It frequently comes up in social science that variables that should be distributed in some way, say normally, end up having a discontinuity in their distribution around certain points. For instance, ...
3
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2answers
2k views

Histogram/distribution fitting for this dataset with unequal and open-ended intervals?

I have this income distribution data for various groups: https://docs.google.com/spreadsheet/ccc?key=0Akwg3n_e05cCdEdtT0VZYU5keW5DVkNoNmpBWmdzeUE As you can see, I have intervals/bins with varying ...
17
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5answers
5k views

Does a univariate random variable's mean always equal the integral of its quantile function?

I just noticed that integrating a univariate random variable's quantile function (inverse cdf) from p=0 to p=1 produces the variable's mean. I haven't heard of this relationship before now, so I'm ...
45
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4answers
82k views

How do you calculate the probability density function of the maximum of a sample of IID uniform random variables?

Given the random variable $$Y = \max(X_1, X_2, \ldots, X_n)$$ where $X_i$ are IID uniform variables, how do I calculate the PDF of $Y$?
4
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3answers
300 views

How to analyse data on metric space?

I just ran an experiment and I'm not sure how best to analyse the data. My data are distance values between objects in a metric space bounded by [0,1]. I have drawn up a probability density estimate ...
0
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1answer
369 views

Cumulative Distribution Function problems

Let $a$ and $b$ be two positive integers. Let also $X$ be a discrete random variable varying in $X(\Omega)=\{1, 2, \ldots, ab\}$ such that for every $x∈X(\Omega),P[X = x]=1/a - 1/b$. a) What ...
3
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0answers
296 views

Goodness-of-fit test without analytical PDF and CDF

I have closed form moment-generating function and characteristic function of a distribution, which describes waiting time of a continuous univariate random process. However, I cannot analytically ...
3
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1answer
820 views

What is the best way to visualize a single numeric variable as a heatmap?

I'm a hobbyist programmer whose friend recently took a business trip overseas. He's polled our mutual friends for bets on the size of his email inbox when he returns. I'd like to visualize this as a ...
1
vote
1answer
1k views

Problem printing greyscale or B&W ggplot2 images

I have an issue with printing/creating black and white and greyscale images with ggplot2. I have been trying to make some black and white, and greyscale graphics for publication, they look perfect on ...
2
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1answer
8k views

What are density and intensity of point pattern?

A simplified explanation with the focus on the following questions is being inquired. Appreciation goes in advance to whom provides scientific--simple--practical explanation. Having an area A ...
4
votes
1answer
196 views

PDF for a function of random variables

If $g=f(x,y)$ is a function of independent random variables $x$ and $y$ then how do we arrive at the expression for the probability density function of $g$, $$f_G(g) = \iint f_X(x)f_Y(y)\delta(g-f(x,...
2
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0answers
94 views

Density related to sparseness measure

Are there any multi-variate continuous distributions whose probability distribution functions give high values for sparse vectors and low values for dense vectors, i. e. indicating the sparseness of ...
1
vote
1answer
1k views

How can I combine histograms with a density plot in R?

Starting with a really simple example, a bi-modal distribution, however, keeping in mind that this would likely be applied to more complex cases where modality is not as apparent, nor limited to 2 ...
1
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0answers
170 views

What's the difference between $p(x \mid \theta)$ and $p(x; \theta)$? [duplicate]

Possible Duplicate: Meaning of probability notations $P(z;d,w)$ and $P(z|d,w)$ Can you tell me the difference between $p(x \mid \theta)$ and $p(x; \theta)$? I know $p(x \mid \theta)$ is the ...
1
vote
3answers
326 views

Expected value of a random variable in a range

I have a random variable $x$ with $E(x) = \mu$ and PDF $f(x)$ and CDF of $F(x)$. Is there any way to represent the $E(x|x<\bar{x})$ in terms of $\mu$ and $f(x)$ or $F(x)$? i.e. to write the ...
2
votes
1answer
1k views

Plotting density in two different ways gives wildly different looking curves

I have a very strange problem. I don't think it's a bug because I've tried it a few ways with similar results. I have three sets of data that are all related to each other. When I make density plots ...
37
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2answers
4k views

Intuitive explanation for density of transformed variable?

Suppose $X$ is a random variable with pdf $f_X(x)$. Then the random variable $Y=X^2$ has the pdf $f_Y(y)=\left\{\begin{array}{ll}\frac{1}{2\sqrt{y}}\left(f_X(\sqrt{y})+f_X(-\sqrt{y})\right) & y \...
15
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2answers
12k views

Area under the “pdf” in kernel density estimation in R

I am trying to use the 'density' function in R to do kernel density estimates. I am having some difficulty interpreting the results and comparing various datasets as it seems the area under the curve ...
7
votes
2answers
5k views

What is the best way to discretize a 1D continuous random variable?

Say I have a 1-dimensional continuous random variable $X$, with PDF $f(X)$, CDF $F(X)$ and inverse CDF $F^{-1}$. What is the best way to discretize $X$? To keep things clear, let $Y$ denote the ...
5
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1answer
1k views

Estimating PDF of continuous distribution from (few) data points

What are some good, established methods for estimating the probability density function (denoted $f(x)$ from here on) of a continuous distribution, given a sample of points $x_1, \ldots, x_n$ drawn ...
5
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2answers
2k views

Best practices for density estimation of discrete & continuous random variables

I am currently trying to estimate the density of a joint distribution of K single dimensional RVs. I have at my disposal a set of N sample points, each of which represents an outcome of the K RVs. ...
2
votes
1answer
893 views

Calculating event probabilities in mixed, discrete/continuous distributions

This is a simple question. I am dealing with a "clipped" normal distribution -- say, $N(0,0.5)$ clipped between $[-1,1]$. I would like to calculate the "probability" of a sample, but I know that in $...
3
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0answers
70 views

Estimating the distribution of a very large population of known size and unknown variance

I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the shape of the underlying distribution. However, I ...
8
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2answers
2k views

Density estimation with a truncated distribution?

I have some data which is clearly truncated on the left. I wish to fit it with a density estimation that will handle it in some way instead of trying to smooth it down. What known methods (as usual, ...
7
votes
2answers
2k views

Generating random samples from a density function

How does a computer algorithm set up to take as input an arbirary bivariate probability density function, generate pairs of numbers from that distribution? I have found a routine called simcontour ...
3
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1answer
439 views

Constrained kernel density estimation

Suppose you are trying to estimate the joint density $p(x,y)$ based on observed $(X,Y)$. However, you know that the marginal density $p(x)$ is uniform. How can you use this information to improve ...
2
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1answer
116 views

Density function question

Suppose $x_{1}, x_{2} \dots x_{N}$ are gaussian RVs with variance $S$ and mean $1$. What is the density function of $$\frac{ |\sum_{n=1}^{N}x_{n}|^{2}}{\sum_{n=1}^{N}|x_{n}|^{2}}\text{?}$$
2
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0answers
203 views

Modeling membership function given some survey data or empirical distribution

For example, I have a set of numbers (say 0 to 10) that are presented to 100 subjects. Each subject is asked whether the number is a small or a large number. The results are that 100 people think ...
21
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1answer
1k views

Marginal distribution of the diagonal of an inverse Wishart distributed matrix

Suppose $X\sim \operatorname{InvWishart}(\nu, \Sigma_0)$. I'm interested in the marginal distribution of the diagonal elements $\operatorname{diag}(X) = (x_{11}, \dots, x_{pp})$. There are a few ...
12
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3answers
5k views

Closed form formula for distribution function including skewness and kurtosis?

Is there such a formula? Given a set of data for which the mean, variance, skewness and kurtosis is known, or can be measured, is there a single formula which can be used to calculate the probability ...
3
votes
1answer
103 views

Given pdf of $I$ and $R$ (both $I$ and $R$ are independent RV's), how to find pdf of $W =I^2\cdot R$?

The title defines the question. May be the concept would do...like how to go about it? Thanks.
14
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3answers
5k views

Best way to put two histograms on same scale?

Let's say I have two distributions I want to compare in detail, i.e. in a way that makes shape, scale and shift easily visible. One good way to do this is to plot a histogram for each distribution, ...
8
votes
2answers
6k views

Problem calculating joint and marginal distribution of two uniform distributions

Suppose we have random variable $X_1$ distributed as $U[0,1]$ and $X_2$ distributed as $U[0,X_1]$, where $U[a,b]$ means uniform distribution in interval $[a,b]$. I was able to compute joint pdf of $(...
7
votes
2answers
4k views

Use Empirical CDF vs Distribution CDF?

NOTE: I purposely did not label the axis due to pending publications. The line colors represent the same data in all three plots. I fitted my data using a negative binomial distribution to generate a ...
10
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5answers
4k views

Generate random multivariate values from empirical data

I'm working on a Monte Carlo function for valuing several assets with partially correlated returns. Currently, I just generate a covariance matrix and feed to the the ...
143
votes
6answers
75k views

Can a probability distribution value exceeding 1 be OK?

On the Wikipedia page about naive Bayes classifiers, there is this line: $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve ...
2
votes
2answers
3k views

How can I display empirical pdf of my 100x1 vector data in Matlab?

I have a data which is 100x1 vector. How can I display its empirical pdf in Matlab? Also, if I want to compare the pdf of three vectors on the same graph, then how to do that? Right now I am using ...
17
votes
4answers
46k views

Difference between histogram and pdf?

If we want to visibly see the distribution of a continuous data, which one among histogram and pdf should be used? What are the differences, not formula wise, between histogram and pdf?
10
votes
2answers
2k views

Kernel bandwidth in Kernel density estimation

I am doing some Kernel density estimation, with a weighted points set (ie., each sample has a weight which is not necessary one), in N dimensions. Also, these samples are just in a metric space (ie., ...
5
votes
3answers
909 views

Series expansion of a density function

Here's something I've wondered about for a while, but haven't been able to discover the correct terminology. Say you have a relatively complicated density function that you suspect might have a close ...
21
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2answers
67k views

Finding the PDF given the CDF

How can I find the PDF (probability density function) of a distribution given the CDF (cumulative distribution function)?