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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24 views

Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?

Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x}, \;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda e^{-\lambda x},\;\; y>0.$ Find the probability ...
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1answer
44 views

Poisson Distribution with variable average rate λ?

When I observe the popular times of a store / place / online shop on Google,† sometimes the bar graph has one peak (maximum) but sometimes it has two (global maximum and a local maximum). I was ...
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16 views

Conditional density estimation review

Can anybody suggest textbooks/papers presenting a relatively up to date literature review on the subject of conditional density estimation? Specifically, I'd be interested in a comprehensive summary ...
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1answer
26 views

Normalizing the joint probability density

I computed the kernel estimators for the copula density for two random variables using: library(kdecopula) kde.fit <- kdecop(u) As the values of density can be greater than one I was wondering ...
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1answer
71 views

What a normal curve actually is?

I am fairly new to Stats and thus this question. I was going through different materials to learn stats. Somewhere it's mentioned that a population, if plotted against frequency on a bar graph can ...
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25 views

Max value of a pdf

We have $f_x(x;μ, σ^2) $ as the pdf of a normally distributed variable $X$. What is the maximum value of the pdf? I thought because the pdf is normally distributed, so it must be the case that mean ...
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1answer
16 views

the variance of a gaussian PDF?

A problem is this: The probability density function of the univariate Gaussian with mean $ μ $ and variance $σ2, N(μ,σ2)$: $$f_x(x) = \frac{1}{\sqrt(2*pi*σ2)} * e^-(x-μ)^2/(2*σ2)$$ The pdf of a ...
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2answers
37 views

Why can I use a PDF when computing bayes rule?

My understanding is that PDFs are 0-valued at all individual points, and only when we integrate over a specific region do we get a non-zero value. However, my professor keeps using PDFs when ...
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1answer
14 views

Confusion over dmultinom function arguments and meaning with respect to Math [closed]

What do the arguments in the dmultinom function mean? ...
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2answers
43 views

get a distribution from a set of values approximating a PDF in R [closed]

I have a set of points: ...
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1answer
26 views

Rician Probability Density Function [duplicate]

In this graph related to the Rician Distribution why on the $y$ axis I see values greater than $1$? It does not mean the probability? Please may I ask how to read this graph and have some information ...
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30 views

F distribution definition with gamma function or square root [closed]

On wikipedia (https://en.wikipedia.org/wiki/F-distribution) I see the pdf of F distribution defined using squared root function, while other places I see the pdf defined with the gamme function. E.g ...
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1answer
14 views

Find joint pdf table of two discrete independent random variables $X$ and $Y$

Given the pdfs of two discrete independent variables $X$ and $Y$, write the joint pdf. There is a property that $ if\ \ p_{XY}(x,y) = p_X(x)p_Y(y) \ \forall i,j \Rightarrow \text{X,Y are ...
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1answer
26 views

Find expected value with pdf and LOTUS

I am currently trying to solve a problem and can't figure it out. I have done this before, but I can't remember all of the details and can't find a reference example. Let's say I have a pdf $$f(x)=\...
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1answer
34 views

example of when the likelihood function does not sum up, or integrate to $1$? [duplicate]

Could someone please give an example of when the likelihood function does not sum up, or integrate to $1$? I have seen this question with the first answer but it really confused me - why are we ...
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0answers
25 views

Inverse transform method with piecewise pdf

I am having trouble using the inverse transform method with the generalized inverse $$F^{-1}(u) = \inf \{x : F(x) \geq u\}$$ In this case, I have a piecewise pdf $$f(x) = \begin{cases}x, & 0 \...
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1answer
48 views

Obtaining cdf from pdf when pdf is defined on limited region/support

This is a very simple question, but I want to make sure I am doing it correctly. I have the pdf from a Pareto distribution: $$f(x) = 160 x^{-6}, \ \ 2 \leq x < \infty$$ and want to obtain the ...
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2answers
44 views

Joint distribution of random vector and a linear combination of it

If $\mathbf{X} \sim \mathcal{N}_n(\mathbf{\mu}, \mathbf{\Sigma})$, what is the joint distribution of $(\mathbf{X}, \sum_{i=1}^n c_i X_i)$ where $c_i$ are constants? I've tried to work this out, but ...
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0answers
28 views

How are probability functions derived? E.g. normal, Poisson, t-distribution

The idea behind PMFs is simple - how likely is a given discrete event to happen? For example, it is intuitive to see why the PMF of a binomial distribution is $ Pr(X=k)= \left( { \begin{array}{} ...
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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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15 views

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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1answer
24 views

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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395 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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33 views

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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1answer
35 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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1answer
56 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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37 views

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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0answers
47 views

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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17 views

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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2answers
99 views

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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1answer
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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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16 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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0answers
19 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
42 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
36 views

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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5answers
2k views

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
31 views

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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2answers
47 views

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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1answer
26 views

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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1answer
41 views

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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0answers
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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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2answers
67 views

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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17 views

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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0answers
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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0answers
13 views

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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0answers
32 views

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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0answers
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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