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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Problem on Clustering Discrete Input using GMM

I want to do clustering using Gaussian Mixture modeling (GMM) on a set of data which is a 5-dimension vector of real values $(x_1,x_2,x_3,x_4,x_5)$. However the clustering result were pretty bad, ...
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Correlation between two variables from muli-dimensional distribution?

Possibly an elementary question, however I can't find an satisfactory answer. If I have a bivariate distribution, then clearly its easy to determine the correlation between those two variables, For ...
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34 views

PDF of multivariable function from known distribution of components [duplicate]

How can I determine the pdf of the following function: $$z(x,y) = \sqrt{ax^2 + by^2}$$ given the constants $a,b$, the means $\mu_{x},\mu_{y}$ and variances $\sigma^2_{x},\sigma^2_{y}$ of the ...
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Adding two distributions with proportional means and standard deviations

I need to know the mean $\mu_z$ and standard deviation $\sigma_z$ of a log normal distribution $Z(\mu_z,\sigma_z)$ which is the sum of two other log normal distributions--$X(\mu_x,\sigma_x)$ and ...
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17 views

Plotting PDF of monthly sales figures in excel

For a given month, I have avg 24,500 sales,and a st dev of 76,000. Since there are many,many entrys of $0, I have standardized each value. How do I go about taking these standardized values and ...
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72 views

One Sigma error and 68% tolerance interval

I have first a clarifying question, and second, a question asking about how to do something, depending on the answer to the first question. Suppose you have a set of data of some PDF which is ...
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7 views

compare multiple size frequency distributions

I have a dataframe with annual size frequency distributions: ...
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1answer
81 views

Problems generating a sample from a custom distribution with log

I'm trying to generate a sample from a family of distributions. In particular I would like to be able to obtain a sample from the survival function: $$1-F(x) = c x^{-a} \log^b(x)$$ with a proper ...
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1answer
64 views

How are percentiles distributed?

I was taking a look at this page, and I can't seem to understand why the frequency plot of the percentiles is uniformly distributed. Distances between percentiles are not equal, so why is the ...
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2answers
42 views

Probability distribution of the magnitude of a circular bivariate random variable?

I'm very new to this topic. I have a distribution similar to the picture below but with the center at zero. As I said, I'm very new to this, but if I understand correctly, if there was no hole in ...
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1answer
43 views

Density forecasting

I am having some troubles obtaining density forecasts of any returns series. As I couldn't find any numerical examples on the Internet, I would like to ask you guys for some help. My goal is to ...
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15 views

Relationship of instantaneous variance and VIX with conditional density functions

I read a paper about the VIX which contains a stochastic volatility model. To estimate the parameters of SV (to be later able to price VIX futures), they derived a conditional probability density ...
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2answers
141 views

How do you find a cutting point / strong slope within one-dimensional data

I have one-dimensional data. I want to find possible natural cutting points (strong slopes) within the data. For instance, if the data is ...
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1answer
21 views

Pattern Recognition - Visualizing Dense Data Points

I have a sample of around 5000 data (2D) points that are generated through a simulation of a cryptocurrency's mining events of following form. In column 2 one can see identical y-values with ...
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2answers
164 views

relation between probability and probability density function

According to this link: PDF To translate the probability density $ρ(x)$ into a probability, imagine that $I_x$ is some small interval around the point $x$. Then, assuming $ρ$ is continuous, the ...
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2answers
79 views

Why is the CDF of a sample uniformly distributed

I read here that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution. I have ...
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Pmf of compound Poisson process

Can I obtain an analytic expression for pmf of compound Poisson process? $Y_t = \sum \limits_{i=1}^{Xt} D_i$ where $X_t$ ~ Poisson($\lambda$) and $D$ ~ Geometric($\rho$)
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38 views

why we use precision instead of variance in prior?

I am new comer in stat field. I am wondering about using precision instead of variance in prior.
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35 views

Probabilities of conditional expectation values in uniform distribution

Let's consider a continuous random variable $X$ as follows: $f_X(x)=\left\{ \begin{array}{ll}\frac{1}{2}, &\mbox{if} \ x\in[0,1] \\ \frac{1}{4}, &\mbox{if}\ x\in(1,3]\end{array}\right.$ ...
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40 views

Why does phase randomizing the Fourier transform of a data set render it Gaussian?

Let's say I have a data set $s_n$. I take the Fourier transform of this data set to obtain $\tilde{s}_n$. I randomize the complex phases of $\tilde{s}_n$ and I take an inverse transform to obtain ...
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164 views

Fastest way to solve Bayes estimator problem

The below problem is from an old PhD qualifying exam in our department. My own solution below is time-consuming and quite possibly wrong. It also relies on recognizing a less common distribution, so I ...
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1answer
28 views

sampling distribution of the mean for arbitrary 1-D pdf

I want to compute the sampling distribution of the mean for $k$ samples from an arbitrary, known probability density $f(x)$, with $x \in \mathbb{R}$. What is the most efficient way to do so ...
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1answer
69 views

Copula density function

In the equation $h_t(x,y|...) = ...$, can anyone explain me why the first derivatives of the marginal distributions are included? $H_t$ is a distribution function and $h_t$ its density function. ...
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3answers
98 views

Difference between density and probability [duplicate]

What is the difference between the density and probability? I have tried R in which I can use both pnorm and dnorm for the ...
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34 views

Why does my multivariate normal have a density greater than 1 (log-likelihood greater than 0)? [duplicate]

I am calculating the log-likelihood of multivariate Gaussian distribution. I am getting a positive log-likelihood. Density function ...
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45 views

Relationship between the Gamma and Beta distributions

I was looking at a proof of the following fact Let $X \sim \mbox{Gamma}(\alpha, 1)$ and $Y \sim \mbox{Gamma}(\beta, 1)$ where the paramaterization is such that $\alpha$ is the shape parameter. Then ...
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1answer
46 views

Proof for the p.d.f of minimum and maximum of a sample

The following is a question from a past paper for one of my university statistical inference modules, and I know how to use the formula for each the max/min, but Assume that the sample $X_1, X_2, ...
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42 views

2D Kernel density estimation with uncertainties

I would like to perform bivariate KDE with Gaussian kernels (preferably using Python, or R) of a dataset with heteroscedastic uncertainties. What would be the correct way to do this: to rescale a ...
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2answers
64 views

How to compute a marginal probability function from a joint probability function?

I am looking at part a) and I have found the marginal p.f for $Y$ to be $e^{-2}2^{y}/y!$. I have set up for the equation for the marginal p.f for $X$ but I have no idea how to start it. Help would ...
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70 views

Is every cumulative probability density function Borel measurable?

I have seemingly simple question, which does not need to have a simple answer :) Is every cumulative probability density function Borel measurable?
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40 views

Imposing a model on a pdf

(This question is an attempt to zoom in on the key issue in this question using as little information as possible.) Lets say I want to derive the likelihood function of $\beta$ given $x$ and $y$ for ...
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44 views

Derivation of likelihood function for latent variable model made explicit

I am trying to make the steps deriving the likelihood function for the following latent variable model as explicit as possible: $$Y^0=X\beta + u$$ where $$u \sim NID(0,\sigma^2).$$ The observed data ...
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Estimate density before or after separating into pre-defined bins?

I want to estimate the density of positive real item price data so that I can predict expected revenue per transaction given a known service fee schedule. Suppose also that the fee schedule is defined ...
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50 views

Probability Density Functions of a Worker and his Waking up routine

The Problem: A worker wakes at 6 am and lies in bed for up to 2 hours. Upon rising it takes him an hour to shower and prepare which is preceded by him doing whatever he pleases. He never leaves for ...
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37 views

How to extrapolate future probability density functions if you have a time series of them as input?

I'm sorry for lack of technical vocabulary, I'm not a mathematician but an undergraduate student in business informatics. This is my current situation: I am given an observations vector ...
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1answer
58 views

Understanding “the kernel has zero mean”

I am trying to understand kernel density estimation and found the graphic below illustrating different kernel functions on Wikipedia. I have no trouble reconciling it with the two statements "the ...
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20 views

Multivariate extension of Welch-Satterthwaite approximation

The Welch-Satterthwaite approximation can be used to approximate the distribution of sums of gamma random variables. See section 4.1 here, for example. Can we use a corresponding approximation for ...
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Additive best-fit parametric kernel estimation?

I'm thinking of a process similar to kernel density estimation, but using fitted kernels. The process would be something like this: Create a histogram $H_1(x)$ with infinitesimal bins (such that the ...
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43 views

How does the stats.gaussian_kde method calcute the pdf?

I am using the scipy.stats.gaussian_kde method from scipy to generate random samples from the data. It works fine! What I have ...
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Interpreting Poisson regression: Deriving results from the CDF

I am trying to interpret a Poisson regression without being very interested in the mean. As a complication, I also have an exposure variable. Let $y_i$ be a count variable, and $p$ the offset, for ...
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30 views

Estimate density dependent on another variable

I have an (unknown) random function $y=f(x)$, i.e. for each value of $x \in [0,1]$ it is a random variable with some distribution. Also I can sample this function, and got values of $y$ for many ...
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2answers
64 views

How is $p_{i}$ for a set of continuous data points related to the probability function $f(x)$?

I have the following set of continuous measurements: 155.08 178 264.81 238 378 140.38 130.5 140.69 155.5 To average this data, I sum the values and divide by the ...
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1answer
111 views

Overlap between two normal pdfs [duplicate]

I have two normally distributed random variables (estimated from two different sets of samples), and I'd like to know how "similar" those variables are (in order to compare the sets). I had the idea ...
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1answer
64 views

How to interpret height of density plot

How should I interpret the height of density plots: For example in the above plot, peak is at about 0.07 at x=18. Can I infer that about 7% of values are around 18? Can I be more specific than ...
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How can I write a skew-normal distribution function given these 3 points?

Suppose I have a set of normally distributed data with mean µ, such that 34% of the data lie between µ–σ and µ, and 34% more of the data lie between µ and µ+σ. Then we know I can write a distribution ...
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1answer
102 views

How to find the normalizing constant for a distribution of unbounded support?

The probability density of a random variable is $$f(x) = ax^2 e^{-kx} ;k\gt0,0\le x\le \infty$$ What is the value of $a$? I understand that first we'll have to take the integral of the function ...
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1answer
176 views

Deriving Density Function (pdf) from Distribution Function (cdf)

A random variable $V$ has the distribution function: $$ F(v) = \begin{cases} 0, & \text{for $v<0$ } \\ 1-(1-v)^A, & \text{for $0\le v\le1$ } \\ 1,& \text{for $v>1$ } \\ \end{cases} ...
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29 views

Why small values produce undulating densities when ploting logarithm of a loguniform prior (in R)?

I am using a program that draws random values in a log-uniform distribution let say between 1 and 100. When I plot the density of the produced values with R it looks like a log-uniform distribution ...
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54 views

What is the name of this distribution family?

I am trying to identify this probability density function so I can read up on it to find confidence intervals for $\theta$: $$f(x;\theta,v)=\frac{\theta v^\theta}{x^{\theta+1}}I_{[v,\infty)]}(x);\ ...
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Is there any general formulation procedure of probability density functions?

There are so many probability density functions for continuous variables around the world. Unlike the probability mass functions of discrete variables, these PDFs do not directly give you the ...