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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Likelihood of a Poisson-described event to occur in the next second

Consider a recurring event for which the time periods between consecutive events is exponentially distributed. For argument's sake, I'm waiting for a taxi on a busy street. How might one calculate the ...
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1answer
30 views

Choosing among PDFs

This is a pretty broad question. I just learned that two random variables can have the same moments but different PDFs. Take $\mathbb{E}[X_i] = \mu$ and $\mathbb{Var}[X_i] = \sigma^2$. Since there are ...
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1answer
23 views

conditional probability, change of variable and Jacobean

I have a question, and I am guessing that the question arises due to my lack of good understanding in the change of variable technique. I would like to evaluate $f_X(x)$. When $f_Y(y)$ exists, I can ...
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2answers
52 views

How to measure similarity of bivariate probability distributions?

I have three different distributions of 2D data: or Now I like to know whether distribution two is more similar to distribution one (2 to 1) than distribution three is to distribution one (3 to ...
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28 views

Accounting for measurement bias using a histogram or violin plot and numerical data in R

The Problem I have a gardener whose job it is to measure trees in meters using a measuring stick. She measures the heights of 1,000 trees. I plot these measurements on a violin plot in R. I am ...
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21 views

Help with t-distribution

I've been trying to show that integral of $\mathbb{R}$ of the pdf of the t-distribution is indeed equal to 1 and have been having trouble. The pdf of the t-distribution, given in my book (Statistical ...
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18 views

When to use CRPS (Continuous Rank Probability Score)? What are the alternatives? What are the advantages and disadvantages?

Please correct me if I'm wrong, crps is new for me. I want to understand it better. We have to minimize crps, which is based on the cumulative distribution function of the data. While the information ...
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1answer
25 views

Can I use glm with Poisson family if counts data are treated as density?

Imagine you have data of birds counted in an area - let's say, you count 18 birds in a surveyed area of 1,3 km^2. Imagine you relate this counts to 1km^2, so that you have 13.9 parrots per km^2. ...
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1answer
10 views

How to compute the pdf for logit/probit models?

According to the probit/logit models, the change in probability due to a change in an explicative variable x is given by the following equation: P(Y = 1 |X) = ...
2
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1answer
29 views

About the burr distribution 3 parameters

In my papers the probability density for a burr distribution is given as $f(x) = \dfrac{\gamma \tau \alpha^{\gamma}x^{\tau - 1} }{(\alpha + x^{\tau})^{\gamma + 1}}$ however i have encountered this ...
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51 views

Where is the maximum bias and variance in a histogram as non-parametric density estimator?

I am a little bit confused about bias and variance of non-parametric density estimators and hope you can help me. Assuming a constant bandwidth and sample size, I am wondering at which points of the ...
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215 views

Why does a Cumulative Distribution Function (CDF) uniquely define a distribution?

I have always been told a CDF is unique however a PDF/PMF is not unique, why is that ? Can you give an example where a PDF/PMF is not unique ?
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95% region from a bivariate density

I have a bivariate data which can be displayed as a scatterplot of which marginal distribution is Uniform(0,1). Using 'kde2d' function in R, I can also obtain two-dimensional density of these data. ...
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1answer
23 views

Consistency of density estimation under marginalization

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be samples from some unknown distribution $p(X,Y)$ and $\hat{p}(X,Y)$, $\hat{p}(Y)$ density estimates of the joint and marginal distributions (i.e., for the estimation ...
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Conditional Density Formula (using Gaussian copula)

Is it true that $R_k$ can be calculated using Pearson correlation, Spearman's $\rho$ or Kendall’s $\tau$? And is it true that If we use Pearson correlation, then the correlation(covariance) $R_k$ in ...
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3answers
52 views

Intuition of pdf of a continuous random variable [duplicate]

What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if ...
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1answer
43 views

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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2answers
68 views

Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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1answer
29 views

marginal conditional distribution from MCMC output [duplicate]

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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1answer
38 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
140 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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2answers
83 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
50 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
26 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
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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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3answers
176 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?
3
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1answer
36 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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1answer
37 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
41 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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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
43 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
288 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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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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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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134 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
109 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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172 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
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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
48 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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1answer
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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
52 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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177 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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79 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
42 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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69 views

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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3answers
124 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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30 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
64 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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151 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 ...