Questions tagged [cumulative-distribution-function]

Cumulative distribution function. While the PDF gives the probability density of each value of a random variable, the CDF (often denoted $F(x)$) gives the probability that the random variable will be less than or equal to a specified value.

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Simulating a joint distribution with the inverse method

I have the following joint distribution: $$f(x, y) = 3x^2y^xe^{-x^3}(1 + x),\quad x \gt 0,\ y \in (0,1).$$ I want to simulate a sample of this distribution through the inverse method but I don't know ...
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How to compute the marginal CDF of a joint density?

I am trying to compute the cumulative distribution function of a random variable $u$ that has the following density: $$f(u) = \int_{1}^\infty \frac{e^{-4uv}}{v^5}dv$$ for $u \gt 0$. What I've tried ...
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Deriving the CDF of student $t$-distribution

I am trying to derive the cumulative density function(cdf) of $t$-distribution, define as in https://en.wikipedia.org/wiki/Student%27s_t-distribution#Cumulative_distribution_function I derive it by ...
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Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability?

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Cumulative Distribution function of ratio of 2 correlated gamma distributed random variables

Let $X \sim \Gamma(k_x, \theta_x)$, $Y\sim \Gamma(k_y, \theta_y)$ be correlated with correlation coefficient $\rho$. The shape and scale parameters are different. i.e. $k_x \neq k_y$ and $\theta_x \...
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Predicting failure (e.g. from weibull) of part that has already survived a certain time

Simple question, but I'm overthinking it. Say that I already have a cumulative failure distribution, cdf, (Weibull, but let's just assume any type of distribution) calculated from historical part data....
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Multivariate Gaussian probability mass inside a sphere

Assume I have some d-dimensional multivariate gaussian $X\sim\mathcal{N}\left(\mu,\Sigma\right)$ and some sphere $C=\left\{ x:\left\Vert x-z\right\Vert_2\le r\right\}\subseteq\mathbb{R}^{d}$. I was ...
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Connection between forms for Generalized Pareto Distribution

On Wikipedia (https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I–IV) one can find the relation between the different types of Pareto Distribution and the Generalized Pareto Distribution (...
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How can I test whether one empirical CDF is to the left or right of another?

I am currently working with a box plot (shown below) that consists of two boxes per value of one of the independent variables (call it $x$). The other independent variable is indicated by the two ...
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Computing Gini coefficient for a 2 parameters density function

I have a random variable $X$ defined by the following the density function, \begin{equation} f_{\theta_1, \theta_2}(x) = \begin{cases} \frac{\theta_1 \theta_2^{\theta_1}}{x^{\theta_1 + 1}}, &...
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Calculating the probability of the total duration of N sequential events with different cdfs describing their duration

Be patient, I am not very skilled with cdf. I seem to have a seemingly simple problem for which I either can't seem to find material about or simply lack the vocabulary for. Given are N sequential ...
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Continuity of a multivariate CDF at a point

I want to show that if $(X_1,X_2,\dots,X_n)$ is an $n$-variate random variable, then its CDF is continuous at a point $\vec a$ iff $$P\left(\bigcup_{k=1}^n\{X_k= a_k, X_j\le a_j\:\forall\:j\ne k\}\...
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Represent $P(x,y)$ with $P(x)$ and $P(y)$, what are the pre-conditions we require?

$x$, $y$ are two random variables. Given density functions $P(x)$ and $P(y)$, can we represent the joint probability density function $P(x,y)$? if not, what are the key conditions we require to derive ...
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Compute P(XY) given P(x) and P(y)

Given two random variables x and y. Their PDFs P(x) and P(y) are known. However, if we do not assume the independence between x and y, how can we represent the Cumulative Distribution Function F(xy) (...
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How to build CDF when there are extreme values

I want to build a CDF for some phenomenon, say $P($storm duration $ D \leq d)$. The particularity of that phenomenon is that it has extreme values. I understand that I can fit some PDF (I have data ...
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Distribution Function from Density Function

I'm guessing there was an error in a Probability and Statistics exam I have recently taken. Let $X$ be a random continuous variable and $f$ a function defined as follows: $ f(x)=\left\{\begin{matrix} ...
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Finding probability of all success for in an order statistic

𝑓(𝑦) = 5𝑦^4; 0 ≤ 𝑦 ≤ 1 A group of 3 friends order small cups of soda, from the soda dispenser. If the 3 small cups are considered a random sample from the dispenser fills, find the probability ...
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Understanding Plackett's singular MVN correlation matrix

I'm trying to follow the paper "A Reduction Formula for Normal Multivariate Integrals" (Plackett, Robin L., 1954) which proposes reduction formulae for calculating the cumulative ...
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Figure of merit for multiple simulations of point patterns

I am having problems understanding how I can evaluate a set of (Monte Carlo) simulations based on randomly distributed points. Assume you simulate a random point pattern in a square and you plot the ...
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How do you do a qq plot in R for a discrete Weibull distribution?

To do this, I think I need to calculate the inverse CDF, but I have learned here that the discrete Weibull (type I) as given by: $$F_{I}(x)=1-\exp\left[-\left(\frac{x+1}{\alpha}\right)^\beta \right]$$ ...
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randomforest for multivariate partial cumulative distribution?

https://cran.r-project.org/web/packages/quantregForest/index.html Give a set of features $X_1, X_2, ..., X_n$ (generated according to some process of uncharacterized distribution) and their values $...
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What is the effect of adding an independent summand on translated quantiles?

Suppose that $Y,Z$ are independent, continuous, non-negative random variables. Suppose also that for some $0<q<1$, $$\tau=\inf\{t>0:F_Y(t)\geqslant q\}$$ and $$\pi = \inf\{t>0:F_{Y+Z}(t)\...
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Do We Need CDF, When We Have A PDF? [duplicate]

CDF is the probability that a random variable takes on a value less than or equal to a fixed $x = a$. Assuming we have a a random variable $X$ that has a PDF, both CDF and PDF have the same ...
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Goodness-of-fit tests for discrete distributions

I have data where only values at large x should fit to a particular distribution whose parameters I wish to determine. I want to do a goodness-of-fit test to find the value of x where the data fit to ...
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Are these descriptions of CDF matching equivalent?

I have two random variables of which I want to transform the empirical CDF of X to the empirical CDF of Y. I don't have a strong ...
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How to derive the distribution of 1 - X [duplicate]

Suppose X is a random variable from the Beta(0.5, 1) density. I want to derive the distribution of Y = 1 − X My attempt: $F_{1-X}(\alpha)= P(1-X \leq \alpha) = P(1-\alpha \leq X) = P(X\geq 1-\alpha) =...
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How to derive the distribution of 1 - X if X is Beta(0.5, 1)

I've been reading about uniform distributions and I'm wondering how statisticians derive these distributions. Lets say we have X, a random variable from the Beta(0.5, 1) density. How could you derive ...
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How to prove $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$

I've been reading about uniform distributions but I can't see how $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$. Is there a way to prove this using the cumulative distribution ...
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stochastic dominant second and first order

I am a bit confused between first order stochastic dominant and second order stochastic dominant can you give me an example which it is second order dominant but not first order stochastic dominant
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Generate random values using an empirical cumulative distribution function

I have a set of data points that I have used to generate my empirical CDF which looks like this (to simplify things I have reduced the number of points for this question but it shouldn't matter): ...
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1 answer
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Generate random values using an empirical cumulative distribution function [duplicate]

I have a set of data points that I have used to generate my empirical CDF which looks like this (to simplify things I have reduced the number of points for this question but it shouldn't matter): ...
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Is the CDF of the Mean always 0.5 for all kind of distributions?

Can we say that the value of the cumulative distribution function at the mean F(X< Mean) is always 0.5 for all kind of distributions (even ones that are not symmetric)?
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How are these distributions same?

I am reading this paper 'Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations' and struggling with understanding the proof of their main claim (in Appendix A). ...
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Finding Cumulative Distribution Functions and merging them

I made up a data set with n=314, mean =14.27854, standard deviation =2.16547 using p <-rnorm(314,14.27854, 2.16547). Now, I want to compare theoretical ...
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How to derive the solution of $F_S(x)=P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)$?

[EDIT] I came across a received signal-to-interference-plus-noise-ratio (SINR), $S$, of a wireless communication system as \begin{align*} S = \frac{\phi|h|^2\rho_1}{1+|g|^{2} \rho _{2} }, \tag{1} \...
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1 answer
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Question about the expression of the CDF for a standard normal distribution [duplicate]

My question is with regards to the second equation. Why is the term in the integral using the variable y? Shouldn't it be z? Because in the first equation, we have the PDF as a function of z. So to ...
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Question about the Solution for Problem 3.7b, Introduction to Probability (Bertsekas, 2nd Edition)

I am currently working on the problems in Introduction to Probability (Bertsekas and Tsitsiklis, 2nd Edition) and one of the problems is as follows: For Problem (b), the final answer is as follows: ...
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1 answer
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Name of a distribution similar to the exponential

for a simulation I'm using the continuous distribution $$F(x)=1-(1+x)e^{-cx} $$ for $x\geq 0$ with $c\geq 1$. Do you know if this distribution has a name?
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Doubt about distribution function of continuous distribution with (Generalized) Pareto tails

In this paper: https://www.sciencedirect.com/science/article/pii/S0167947315003163 They proposed an estimation method for the parameters of the Generalized Pareto Distribution. Defining: $$ F_n(x) \...
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3 votes
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Why is the Empirical Distribution based on the Cumulative Distribution?

Why is the Empirical Probability Distribution Function based on the Cumulative Probability Distribution Function? I have always seen the Empirical Distribution Function to have a "staircase"...
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Expansion of CDF of normal distribution using integration by parts

How does the author express last F(x) mathematical expression in terms of second last H(x) mathematical expression
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How to Determine Distribution of Dataset? [duplicate]

I am working on iris dataset to compute the CDF, PDF of sepal length (or any other variable). For that, I want to know what distribution followed by sepal variables. Is there any way to determine ...
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What can we say about the distribution of $F(X)$ when $F$ is not invertible? [duplicate]

Let us say $X$ is a random variable with cdf $F$. I know that when $F$ is invertible then $F(X)$ has unif$(0,1)$ distribution. The proof goes like $$P[U\le u] = P[F(X)\le u] = P[X\le F^{-1}(u)] = F(F^{...
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Understanding Simple Inverse Distribution

I know this is quite basic, but I fail to see where my mistake with the following simple example from wikipedia is. \begin{align*} G(y) &= \Pr(Y \leq y) \\ &= \Pr \left (X \geq \frac{1}{y} \...
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Is there a simple method to produce a histogram of the Wilson distribution? [closed]

The Wilson score for some observed binomial distribution $(\hat{p},N)$ gives you a confidence interval on $p$. However, I need a histogram over the possible values of $p$. (With a relatively low ...
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7 votes
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Sklar’s Extension Theorem and support restrictions

This question is about an application of the Sklar's Extension Theorem, whose proof can be found in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look ...
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Examples of Defective Distributions

I was reading about Stochastic Convergence and I came across a term called Defective Distribution. Essentially what they refer to as a {Defective Distribution is a distribution that has all the ...
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2 votes
3 answers
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Kolmogorov Smirnov Test : CDF vs PDF

Does anyone know that why in the Kolmogorov-Smirnov Test, the empirical distribution function is compared with the cumulative distribution function and not the probability distribution function? Is ...
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2 votes
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Numerical evaluation of infinite sums

I am working with Skellam random variables and I would like to evaluate the CDF of the absolute value of a Skellam random variable in which both Poisson random variables have the same rate, $\lambda_1 ...
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Approximate binomial dist in sql

What is the best way of approximating a binomial distribution, given I have the following functions available: normal_cdf ...
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