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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7 views

modelling Cumulative Distribution of a variable with many zeros and few large values [closed]

I have a data set with many zero values and a few large values, making the dataset highly skewed. I wonder, any probable distribution function, that can be fitted to the data, probably any mixture ...
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19 views

integration of log of pdf

Is there any standard name for following integration we know that $\int p(x) = 1$ but what about $\int \log p(x) dx $ ? more clearly, assume that $X \sim P$ and support of $X$ is $[0,1]$ $\int_{0}^{...
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What does division of two multivariate pdf mean? with example

I have come across this example in the book, but not entirely confident what does the ratio means.
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What is the analog of the PDF and CDF for the likelihood function?

In probability, we can find the cdf using the pdf and vise-versa. Integrating pdf yields the cdf. Does integrating the likelihood function yield any important thing? In statistics, $\mathcal{L} (M\...
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323 views

Which pdf to choose for the prior of an angle?

I have a system in which one uncertain variable is a direction in two dimensions. If I want to define a prior for this, is there an elegant way to reflect the fact that the parameter space dimension ...
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What methods are there for estimating distributions based on histograms?

I recently worked on a consulting project where a client wanted to estimate gamma and weibull distributions based purely on histograms rather than raw-data. I have never worked with problems like that ...
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34 views

How do I convert a standard normal RV to a generalised error RV?

The generalised error distribution (sometimes called the generalised normal distribution) is a generalisation of the normal distribution allowing variation of the kurtosis away from mesokurtosis. ...
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55 views

Find the CDF and use it to find all the medians

Show that for every $p$, $0\leq p\leq 1$, the function $f(x)$ = $p*sin(x) +(1-p)*cos(x)$, $0\leq x \leq \pi/2 $, and $f(x)=0$ otherwise, is a density function. Find its CDF and use it to find all the ...
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Can we sample from the wrapped normal distribution and evaluate the density of the sample simultaneously?

In a computer program (written in C++), given $x\in[0,1)$ and $\sigma>0$, I need to sample $y$ from the wrapped normal distribution $\mathcal W_{x,\:\sigma^2}$ with mean $x$ and variance $\sigma^2$ ...
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13 views

Density function of a stationary processus [duplicate]

I have a question: Let X be a wide-sense stationary processes. is a density function of X is almost periodic? why? A continuous function $F:R \rightarrow L_2$ is said to be almost periodic, if for ...
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Predictive density via LOOCV

I am looking for a way to generate a density prediction (in contrast to a point prediction or a prediction interval) in a multiple regression setting without relying on stringent parametric ...
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Request for Explanation: Deriving Probability Density Function of a Maximum Likelihood Estimator of a Uniform Distribution [duplicate]

I am reviewing some practice problems and have both a question and its solution but am struggling to understand them and am hoping someone can help me. I am struggling to follow the logic for ...
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Mean and variance of a non-standard pdf

I have tried to compute the variance and the mean for $\mu=0.5$ of the following PDF using Wolfram cloud but I failed $$ F(z,\mu,\sigma)=\frac{2 (z-\sigma )^2 \exp \left(-\frac{(z-\sigma )^2 \...
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How to derive a pdf of Complete Sufficient Statistic of exponential family

While studying Mathematical statistics through "Introduction to Mathematical Statistics 7th" (by Hogg and Craig), I've been stuck in the Theorem above. The answer of the exercise 7.5.8 is not given in ...
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15 views

Validity of Monte-Carlo method to estimate a probability distribution which follows a power law

I am using a Monte-Carlo method to estimate a probability distribution function (pdf). Basically, I have several input parameters following known distributions, from which I can draw samples, that I ...
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18 views

Mean of PDF in Kernel Density Estimation using Python sklearn

I am using Kernel Density estimation to find the PDF of demand for a product using historical data. I am using the numpy library. Here is the code below ...
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1answer
40 views

How to graph distribution of Order statistics?

Is there a software that can graph the pdfs and Cds of an arbitrary number of order statistics or is there some code such software? How to do it? I'm trying to understand the distribution of order ...
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1answer
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Distribution of sample maximum from exponential distribution

$x_1,x_2,...,x_n \sim \exp(\mu=1)$ where $x_i$ are independently identically distributed. What is the distribution of $z_n = max(x_1,x_2,...,x_n)-\ln(n)$? Below is my work , I am uncertain whether it ...
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What is the PDF for the minimum difference between a random number and a set of random numbers

I have a list (lets call it $ \{L_N\} $) of N random numbers $R\in(0,1)$ (chosen from a uniform distribution). Next, I roll another random number from the same distribution (let's call this number "b")...
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1answer
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Divide beta from a gamma distribution to get another gamma distribution?

In the textbook, there's a distribution like the following, $S=\sum_{i=}^{200}X_i\sim Gamma(\alpha = 200, \beta)$ then the textbook define a new function $P$ obtained by diving the $\beta$, so ...
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1answer
20 views

Sampling posterior distribution of a function

I have the following problem: let's say I have a function $y=f(x)$. Let $f$ be defined for all $x$ but it it might not be invertible. Further assume $x \sim p(x)$ with some probability density $p(x)$. ...
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Why is the probability mass function of a transformed discrete random variable summed over the inverse values of the function?

Let $X$ be a random discrete variable with probability mass function (pmf) of $p_X(x) = P(X = x)$. Let $Y = g(X)$ (from $\mathbb{R}$ to $\mathbb{R}$). Then, why is it that: $$p_Y(y) = \sum_{x \in g^{-...
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Questions about pmf of multinomial distribution with indicator variables

I was reading through the textbook of Introduction to Machine Learning, it introduces $z^{t} = (z_{1}^{t},...,z_{k}^{t})$, where $z_{i}$ is an indicator variable, each with probability $\pi_{i}$, then ...
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Simplifying modified Bessel function of the first kind

The modified Bessel function of the first kind shows up in the normalizing constant of a lot of random variables (e.g. the normal product distribution, the noncentral chi-square distribution, the ...
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Does the below Plot refer to a Valid PDF?

I have tried to creat a valid PDF according to some special function I have got the below Plot , I got it for n=6 Integrand Range[n/n], Now My question here Is : Is that below Plot refer to a valid ...
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GANs: What does the pdf of the sample data p(x) mean? [closed]

In the context of GANs, the concept of a probability distribution comes up as the generator tries to emulate the "distribution" of the data: $p_{data}(x)$. For me, the use of "distribution" here ...
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A Multinomial Geometric distribution? What is this distribution called?

In a problem I am dealing with I repeatedly interact with a distribution of the following form, $$p(n_1, n_2, \ldots n_M)=\binom{\sum_{i=1}^Mn_i}{n_1\cdots n_m} w_0\prod_{i=1}^Mw_i^{n_i}$$ where $p$ ...
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21 views

problem in understand exponential PDF [duplicate]

I'm studying a paper called "Optimization based on bacterial chemotaxis". As it can be understood from its name, it has proposed an optimization algorithm based on the reaction of a bacterium toward ...
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Calculating the truncated version of the squared hyperbolic secant PDF

$ \newcommand{\sech}{\mathop{\rm sech}\nolimits} $ Hello, I have the following Probability Density Function (PDF): $f(x)=\frac{1}{2s}(\sech\frac{x}{s})^2$ This PDF has support for $x\in(-\...
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Spectral Density of a Stationary, Isotropic Gaussian Kernel

I intend to perform a simulation of a Gaussian process. To that end, I use a stationary, isotropic Gaussian covariance function (aka Gaussian kernel, or squared exponential kernel), $k(r)= \exp (-\...
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38 views

Finding distribution of $\frac{X^2}{Y}$ given joint pdf of $X$ and $Y$

So I have the joint distribution of $X$ and $Y$: \begin{align} f(x,y) = cx, \ 0 < x^2 < y < \sqrt{x} < 1 \end{align} and I want to find the distribution of $X^2/Y$. So I set $U = X^2/Y$ ...
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1answer
59 views

Why density plot tails are beyond maximum and minimum values?

I am trying to interpret the tails of a density curve, which go beyond xlims(0 in this case). I understand that area under the curve between any two points represents the probability of that event. ...
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38 views

How to calculate the PDF of the 'difference' between two Beta distributions?

I start with two Beta distributions: $$\mathrm{Beta_A}(p; \alpha_A, \beta_A) = \frac{p^{\alpha_A-1}\,(1-p)^{\beta_A-1}}{\mathrm{B}(\alpha_A, \beta_A)}$$ $$\mathrm{Beta_B}(p; \alpha_B, \beta_B) = \...
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Given distribution of $X$ and $X|Y=y$, is it possible to find distribution of $Y$?

What the title says! My intuition is NO since in Bayesian statistics we typically specify the prior and likelihood, and from those two we can compute the posterior and so on. We can interpret $Y$ = ...
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1answer
27 views

How do you calculate the expectation without a pdf in the context of Center limit theorem, variance

Given Problem 1) To get the variance and covariance the following steps are taken: In the step below to calculate E[Z^2] how do we approach this without a known pdf? For completeness would finding E[...
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1answer
27 views

Is this a valid pdf

$f_x (x)= x$ if $x \in [0,1]$ and $f_x = 0$ otherwise. Is this a valid pdf? It seems to me it is not since the area under the pdf is 0.5.
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Calculation of joint PDF

we have the joint PDF of two RVs $X$ and $Y.$ we also have two RVs $U = f(X,Y)$ and $V = g(X,Y),$ where $f$ and $g$ are two variable functions. How can I calculate the joint PDF of $U$ and $V$? for ...
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Conditional and density probability (normal distribution)

I am trying to solve the following problem: Suppose that $\mu\sim N(1,4)$ and $Y|\mu\sim N(\mu,1)$. Show that: $$\begin{bmatrix}Y \\ \mu \end{bmatrix} \sim N\bigg(\begin{bmatrix}1 \\ 1 \end{bmatrix},...
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How can I calculate this PDF

I'm trying to reproduce the results of a ray tracing paper which uses reinforcement learning. I asked my question in the computer graphics community of this site, but I think my problem can easily be ...
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1answer
38 views

Sum of two continuous random variables

Let R1 and R2 be two independent random variables, both with uniform density at the interval (0,2). What is the probability of R1>1 given that R1 +R2<2? -- What I've tried: I know that $$ P(R1&...
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1answer
46 views

Find conditional pdf given joint

Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$. What is the marginal cdf of $X$? of $Y$? Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
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1answer
21 views

getting a density function from a sample of 500

I'm pretty new to statistics so please excuse me if the answer is obvious. The scenario is the following: I am using mcmc to sample from a posterior distribution of a parameter. I then need to ...
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Mixing probabilities and probability densities

I'm currently working on a Bayesian network designed to find the probabilities for various lung diseases. In the network there are, among others, a normally distributed random variable (body ...
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Gaussian Distribution [duplicate]

Assume we have two continuous Normal RV "X" and "Y". how can I show the conditional PDF f(X|Y) and f(Y|X) is Normal?
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Given a pmf, how is it possible to calculate the cdf?

Given a pmf (probability mass function) for X (random variable): \begin{array}{|c|c|c|c|c|}\hline x&1&2&3&4\\ \hline p(x)&0.4&0.3&0.2&0.1\\ \hline \end{array} How ...
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How to obtain value range of empirical PDF, given the mode and area?

Given an empirical PDF of a continuous random variable $X$, then integrating over its entire defined domain will yield an area of size 1. To find the probability of $X \ge x_1 \land X \le x_2 $ (as in ...
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how to find domain of marginal pdf when its two variables domain are dependent

I have a pdf $f(x,y)=1/π, 0< x^2+ y^2 <1$; 0, e.w. Here, we can see $-\sqrt{1-x^2} < y < \sqrt{1-x^2}$ So, the marginal pdf of $X$ is $$\int_{-\sqrt{1-x^2}}^\sqrt{1-x^2} 1/πy \, dy\,.$$ ...
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Example: Writing the joint PDF $f(x, y)$ as the product of a marginal and a conditional probability function

I am presented with the following notes on Bivariate distribtions: If we can write the joint probability density function $f(x, y)$ of a pair of random variables $(X, Y)$ as the product of a ...
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55 views

Behavior of kernel density estimation

Consider the random variable $X=YZ$, where $Y\sim\text{Normal}(0,1)$ and $Z\sim\text{log-Normal}(0,1)$ are independent. I wanted to assess the accuracy of kernel density estimates for the density ...