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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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How to approach the calculations of probabilities in high dimensions?

I don't really have too much trouble finding probabilities using joint probability density functions (PDFs) (of two variables) by drawing the area of support in the $xy$-plane, and then integrating ...
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+50

Approximate density from moments and quantiles, then sample from it

Situation I need to send R code to a third party to run estimations for me (I will not be able to work with the data directly). I want to simulate data to test some of the estimators before sending ...
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wigner semi-circle distribution random numbers generation

I am trying to generate random numbers in Wigner semi-circle distribution. Since this one does not have the analytical solution for the inverse function of the pdf. I wonder if anyone familiar with a ...
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1answer
31 views

Expressing as a probability density function [closed]

The measuring error x is a normal random variable. Variance of the error = 4. If distribution of x can be shown by a probability density function f(x), how would you find the analytical expression of ...
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1answer
20 views

How to find emprical PDF by using the normalized histogram?

first of all, thank you for your time, here is my question; Is it possible to find emprical PDF by using normalized histogram? I am trying to learn discrete event simulation and what I see is there ...
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Mapping a range of values such that the resulting distribution is uniform [duplicate]

I have a set of values. Let's call the set X with values ... . Those values in [0, 1] have a non uniform distribution (empirically measured). I would like to re-map those values on [0, 1] such that ...
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2answers
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Find mgf from joint pmf

The joint pmf of random variables $ X$ and $ Y$ is given by $$p_{XY}(x,y)= \begin{align} & \frac{e^{-2}}{x! (y-x)!}\quad\text{if}\,\,\,x= 0,1,...y,\ y=0,1,... \\ \end{align} $$ Find its mgf. \...
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1answer
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An approximation to the cdf of the normal from a pdf?

In this paper (p. 36), authors wrote $$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$ Bellow we will use the approximation $$p(n,T) =...
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1answer
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Why is sample mean minus location parameter of shifted exponential gamma distributed?

My book says the following Suppose $X_{i} \sim$ iid $Exp(1,\eta)$ Where $Exp(\theta,\eta)$ is the shifted exponential ie has density $$\frac{1}{\theta}e^\frac{-(x-\eta)}{\theta}$$ for $x \ge \eta$ ...
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Variance Estimator for Density

Is there an estimator to predict the variance matrix of a 2D distribution given the value of its density sampled on a regular finite grid? I'm not even sure that estimator is the right word to use ...
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2answers
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What is this “phenomenon” called?

Below is a histogram of some data, the bins are integers the other parameters are irrelevant. As you can see there seems to be two separate but overlapping normal distributions for odd and even ...
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1answer
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Simulation: Generate random numbers that cluster around an average? [closed]

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
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1answer
51 views

Find $P(X<2Y)$ of a bivariate distribution

$f(x,y)=\frac{1}{2 \pi }\exp(-\sqrt{x^{2}+y^{2}})$ where $x,y$ in $\Bbb R$ My attempt: $$\Bbb P(X<2Y) = \int_{-\infty}^\infty \int_{-\infty}^{2y}\frac{1}{2 \pi }\exp(-\sqrt{x^{2}+y^{2}})\text{d}x\...
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28 views

Reparametrization trick for VAE, prooving that the resulting vector follows a normal distribution

So I've been reading about Variational AutoEncoders and I'm stuck on a little exercise meant to help understand the reparameterization trick. Z is a random vector of K elements with a distribution $q(...
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1answer
21 views

Probability question in Mat

My teacher give me this question: Using MATLAB, generate 10000 Random Vectors of size 500 with the PDF of Gamma distribution. Find the PDF of maximum and minimum of the generated Random vectors. (Use ...
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Substitution for unknown true density in 'Density Estimation Trees'

I'm having a hard time understanding parts of the derivation of the objective function for Density Estimation Trees (reference below) regarding the loss function. Taken from the article (Sec. 3.1): ...
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1answer
37 views

Little problem calculating hazard function

I'm starting to study maths again after a long time without having touch them and I'm currently with survival analysis. I want to get a hazard function h(x), and I know that it can be calculated as ...
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0answers
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Estimate unit record level data from data in bins/histogram/with interval censoring

It's very common for data to be grouped into categories by data providers to preserve confidentiality. For some purposes, for example estimating a median value or for experimental modelling, I often ...
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1answer
33 views

Problems calculating and plotting distribution function

After a long time without having touch anything related to maths or statistics, I decided to give myself another chance. I am currently refreshing some concepts of density and distribution functions, ...
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0answers
75 views

fitting non-normal multivariate distributions in R

I have many (n=317,823) observations on two variables. I want to fit a bivariate distribution to my observations, in order to identify descriptive features of the distribution (quantiles). However, my ...
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2answers
88 views

Fitting pmf of a scaled Poisson distribution and Python histogram plotting

I have a nuclei meanlife of $550\mu s$, for which I've taken the frequency(rate) to be $1/meanlife = 1818$. I then sampled randomly from a poisson distribution with that frequency, taking the ...
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1answer
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Setting boundaries for calculating $P(Y/X>2)$ choosing $dx/dy$ order [duplicate]

Given two independent variables $X$ and $Y$, with marginal pdfs $f_X(x)=2x, 0 \le x \le 1$ and $f_Y(y)=1, 0 \le y \le 1$, calculate $P(\frac{Y}{X} > 2)$. So this can be written as $P(Y>2X)$, ...
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1answer
51 views

Finding the joint CDF using the joint PDF; why can't I do this?

Find the joint CDF of the independent random variables $X$ and $Y$, where $f_x(x)=x/2, 0\le x \le 2, $ and $f_Y(y)=2y, 0 \le y \le 1$. To do this, we can find the CDF separately for each of the ...
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3answers
73 views

Compute $P(Y<3X)$ using joint PDF

I'm given a joint pdf $f_{X,Y}(x,y)=2e^{-x-y}, 0<x<y, 0<y $ and asked to compute $P(Y<3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this ...
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1answer
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Order Statistics; Finding the probability that the first sample is < 0.6, and the last sample is > 0.6

Here is the problem statement below: A random sample of size 5 is drawn from the pdf $f_Y(y)=2y, 0\le y \le1$. Calculate $P(Y_1^{'} < 0.6 < Y_5^{'})$. Here, using formulas for order ...
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21 views

bandwidth setting for density comparison

I intend to compare a list of density distributions. Following one published paper (with similar type of data and same objective), I learned that I have to use Gaussian kernel density estimator, and a ...
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1answer
16 views

Normalizing two independent weights in order to produce output between 0 and 1

I have two scores, alpha and beta, ranging both between 0 and 1. I want to weight these with weight_one, weight_two in order to favour one of these scores over the other. Then, afterwards, I want to ...
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0answers
47 views

Formulae for density function replacing probabilistic function

I am working with markov chains and in specific I am working with (Langrock, Roland MacDonald, Iain L. Zucchini, W (2016)). They define a forecast distribution given as: $$ Pr(X_{T+h}=x|X^{(T)}=x^{(...
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2answers
74 views

Condition Probability - What am I doing wrong?

I am reviewing some notes of mine and refreshing myself with some statistics and I came across a problem that asks me to calculate $P(X>1|Y>1)$ for the random variables $X$ and $Y$ whose joint ...
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1answer
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Proof of direct proportionality between hazard rate function and probability density function

I'm reading up on reliability and I came across this question: Show that if the hazard function is decreasing, the PDF, $f(t)$, is also a decreasing function and its mode must therefore occur at t =...
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1answer
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Finding the CDF given marginal PDF's; setting bounds

In this question, I'm having a hard time understanding how specifically to set the bounds for the CDF. Let $X$ and $Y$ be independent variables. Find the CDF of $W=Y/X$ using the marginal PDFs ...
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0answers
43 views

PDF transformation for y=|x|

Suppose I have the random variable X with a pdf: $$f(x)=exp(-(x+1)) u(x+1)$$ where u is the unit step function; such that u = 0 for x<-1 and u=1 for x>-1 $$y= |x|$$ for $$-1<x<1$$ ...
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1answer
133 views

Limits of a density function

If the limit of a density function exists does it the follow that it is zero? To put it formally $$\exists a \in \mathbb R \lim_{t \rightarrow \infty} f(t) = a \Rightarrow a= 0.$$
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Terminology Confusion: Probability, Likelihood, AND?

currently I have a terminology issue to accurately write a text. I read up on the definitions on probability and likelihood. Given some continuous random variable(s) as far as I can infer, probability ...
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2answers
23 views

Sample Mean expressed using Standard Normal Distribution

Let $\bar{X}_n = X_1 + \dots X_n$ where $X_i \sim N(0,1)$. We can easily verify that $\bar{X}_n \sim N(0, 1/n)$. Thus $\text{Var}(\bar{X}_n) = 1/n$. Let the density of $X \sim N(0,1)$ be denoted $\...
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PDF transformation for many to one function [duplicate]

I would like to find the PDF of the random variable $Y$ given the PDF of $x$. $$Y=sin(x)$$ $$f(x) = 2x/(pi^2) for 0<x<pi$$ and 0 otherwise. Following the tips in the question here: https://...
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1answer
23 views

Calculate the probability of any one independent scenario occurring

I have $3$ independent scenarios and I need to calculate the probability of any one of them occurring. I’ve calculated the probability of the $i$-th event as: : $1-\big(\frac{70000-1}{70000}\big)^1 ...
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2answers
116 views

How to modify the mean and variance/dispersion of a given distribution

I am trying to find a parametric adjustment that allows modifying the mean and variance/dispersion of a given distribution. Ideally, this adjustment would be implemented through a parametric function ...
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0answers
13 views

Conditioning the probability obtained from a machine learning model

I have developed a random forest classifier to predict whether a customer will churn. The data used to produce this model has the following form ...
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1answer
45 views

Proof for simulation of NHPP by thinning

Background: I'm trying to show equivalency between the density function for a non-homogenous exponential process (NHEP?), (i.e. the arrival times of events generated by a non-homogenous Poisson ...
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1answer
50 views

PMF and independence with two discrete random variables?

Each of n people (whom we label 1, 2, . . . , n) are randomly and independently assigned a number from the set {1, 2, 3, . . . , 365} according to the uniform distribution. We will call this number ...
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0answers
28 views

Simulating from an Epanechnikov kernel density estimate in MATLAB / exact form of the Epanechnikov kernel in MATLAB?

It's my first time posting, so apologies if I'm breaking any etiquette. I've used MATLAB's ksdensity function to estimate a density using the Epanechnikov kernel and would now like to make repeated ...
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0answers
40 views

Finding a critical region for a simple test, with PDF $f(x) = (\frac{x}{\theta} + \frac{1}{2}) \space \mathbb{1}_{(-1,1)} (x)$

I'm dealing with a simple inference problem which involve a PDF I've never dealt with before. We have $X_1, ..., X_n$ iid variables where $X_i$ has a PDF $$f(x) = (\frac{x}{\theta} + \frac{1}{2}) \...
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106 views

Calculating a Confidence Interval for a Proportion for a Sample of Different Size

I'm interested in a (preferably analytic) solution or approximation to the following problem: Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
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0answers
43 views

Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
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0answers
14 views

Other than linear transformations of any one Bernoulli distribution, what popular distributions only take two possible values?

I suspect that this may be a silly question, as some sort of isomorphism ought to exist between linear transformations of the Bernoulli distribution and any other distribution that can only take two ...
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1answer
44 views

The probability density function of half-chi-square distribution

Let $X$ be a random variable from a chi-square distribution with 1 degree of freedom. The probability density function (pdf) of $X$ is $f(x) = \frac{\exp{(-x/2)}}{\sqrt{2\pi x}}$, $x>0$. In the ...
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1answer
45 views

Expectation of $h \circ X$

I'm only starting to learn statistics. The definition I've been given for the expected value (expectation) of a continuous random variable X with probability density function (PDF) $f_X$ is the ...
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55 views

Viewing PMF as an instance of a PDF

I'm having difficulties in thinking about the probability mass function (PMF) as a special case of the probability density function (PDF). I understand that PMF's are used in discrete examples, but ...
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Probability density function of a data point given the locations of its four nearest neighbors

The goal is to find the probability density function $p(\mathbf{x} | \mathbf{c}_1,\mathbf{c}_2,\mathbf{c}_3,\mathbf{c}_4)$. Here $\mathbf{x},\mathbf{c}_i \in \mathbb{R}^d$. $\mathbf{c}_1,\mathbf{c}_2,\...