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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Linear transformation of a random variable by a tall rectangular matrix

Let's say we have a random vector $\vec{X} \in \mathbb{R}^n$, drawn from a distribution with probability density function $f_\vec{X}(\vec{x})$. If we linearly transform it by a full-rank $n \times n$ ...
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1answer
16 views

Mean Preserving PDF Spreading

I have a univariate discrete random variable and a histogram representing its PDF (which is asymmetrical). Is there a known way to increase/decrease the variance of the distribution (i.e. scaling it ...
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4 views

Find the probability of a new data point using “density” function in R [migrated]

I am trying to find the best PDF of a data that has Gaussian distribution, using the "density" function in R. Now, given a new data point, I want to find the probability of this data point based on ...
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1answer
28 views

marginal conditional distribution from MCMC output

I have a MCMC sampler that targets $$\mathbb{P}(U_1,U_2,...U_n \mid G(U) \leq 0)$$ where $U=(U_1,U_2,...U_n)^T$. I realize now I am more interested in estimating the conditional density $$p_k = p(u_k ...
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23 views

Density Function Estimation

Given a sample of $n$ observations, which are assumed to be $i.i.d.$ and generated from a continuous probability law. Consider the question of estimating the density function $f(x)$. There are two ...
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1answer
85 views

Show that $\min(U,1-U)$ and that $\max(U,1-U)$ are uniform

Let $U$ be uniform on $(0,\ 1)$. Show that $\min(U,\ 1-U)$ is uniform on $(0,\ 1/2)$ and that $\max(U,\ 1-U)$ is uniform on $(1/2,\ 1)$. I'm not sure how to approach... the only hint i have is that a ...
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72 views

Finding the Mean and Variance from PDF

A random variable $n$ can be represented by its PDF $$p(n) = \frac{(\theta - 1) y^{\theta-1} n}{ (n^2 + y^2)^{(\theta+1)/2}}.$$ $\theta$ is a positive integer and $y$ is a positive parameter. If ...
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“The total area underneath a probability density function is 1” - relative to what?

Conceptually I grasp the meaning of the phrase "the total area underneath a PDF is 1". It should mean that the chances of the outcome being in the total interval of possibilities is 100%. But I ...
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1answer
46 views

Get probability distribution function from density function

For a given density function, how does one find its distribution function? For example, I have a density function: $f(x)= \begin{cases} t ^2 / 9 & \text{if } t \in (0,3)\\ 0 ...
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1answer
19 views

Interpretation of cartesian product of the support of marginal distribution

Suppose we have a multivariate data set, $s = (s_1, s_2, ... s_p)$ and each $s_i$ is distributed with a distribution that has finite support (we'll call each $s_i$ a "source"). Let us denote the ...
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1answer
18 views

On transformations of random variables, discrete vs continuous [duplicate]

Suppose we have a discrete r.v. $X$, take $Y = g(X) $ where $g$ is one-to-one and onto- If we want to obtain the new pdf for the discrete r.v. we simply notice that $$f_Y(y) = P(Y=y) = f_X(g^{-1}(y)) ...
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164 views

The probability of a random variable being larger than a sequence of random values

Suppose we have a fixed, known, $n$, and each $x_1 \ldots x_n$ is a random number generated uniformly over $[0,1]$. What is the probability that $x_n$ is the largest value in the sequence?
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1answer
35 views

PDF of mixture of random variables that are not necessarily independent

I am trying to derive the expression for the PDF of a weighted mixture of n random variables. Let us taken $n=3$ and define $$X = \alpha_1 S_1 + \alpha_2 S_2 + \alpha_3 S_3$$ $$E[X^2] = 1$$ $s_1$, ...
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14 views

Simulate birth with Rstudio - draw pmf and cdf [migrated]

I have a little exercise to solve with Rstudio for my statistics exam. I tryed to translate it in english, so if something isn't clear please ask me for explanations. "Simulate 100,000 births and use ...
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1answer
36 views

question about a Rosenthal inequality

What is the usefulness of Rosenthal inequalities in (kernel) density estimation where $\xi _i .... \xi _n$ are independent random variables, $\mathbb{E}\xi_{i} =0$ and $c(p)=15p/lnp$ for $p>2$ ...
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1answer
39 views

Compare two unnormalized density functions given the values at samples

I have the density values of two unnormalized density functions $p$ and $q$ at 2000 points: $\mathbf{p} = (p(x_1), p(x_2), \dots p(x_{2000}))$, $\mathbf{q} = (q(x_1), q(x_2), \dots q(x_{2000}))$. Now ...
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24 views

How to modeling the movement of an object? [closed]

I have implemented the condensation algorithm in order to track a moving object in video sequences, so I would improve the predictive step. Currently the state includes only the coordinates of the ...
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1answer
37 views

Update Rules in Expectation Maximization

I am emulating a certain PDF behaviour using a function. However, due to divergent improper integral, I don't have a closed form expression for the normalization constant. To get the PDF, I just ...
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1answer
63 views

Find the mode of a probability distribution function

I am trying the find mode of a probability distribution function given by \begin{equation} g(x/\alpha,\beta,\sigma)=\frac{1}{\Gamma \left( \alpha ...
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23 views

Geometric construction of copula - question regarding C-volume

I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: ...
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13 views

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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2answers
110 views

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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1answer
99 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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165 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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1answer
30 views

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
32 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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55 views

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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1answer
27 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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1answer
50 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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166 views

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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44 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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1answer
33 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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118 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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26 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 ...
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1answer
60 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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1answer
134 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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36 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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1answer
68 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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48 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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38 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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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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91 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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54 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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1answer
58 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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73 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 ...