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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How to fit a function to a CDF in R?

Background: I've been given a dataframe that contains data for a CDF. The column X contains the 250 $X$ values, and the column P ...
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Asymptotics of the survival function for Anderson Darling distribution?

I am using the ADinf procedure of Marsaglia & Marsaglia to compute the CDF of the Anderson Darling statistic. I am interested in the survival function, 1 minus ...
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CDF that combines properties of Pareto and Exponential

Let $Y$ be a random variable defined on the domain $[1;\infty)$ that is distributed according to the cdf $G_Y(y)$. A Pareto distribution, $$ G_Y(y) = 1 - y^{-\theta}$$ has the property that $$ P(Y&...
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How can I convert kernel quantiles into sample quantiles?

I calculated the quantiles for an Epanechnikov kernel which I'm using to estimate the density of a sample. What I need is to find the sample quantiles knowing that it is composed of many Epanechnikov ...
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181 views

CDF*[1-CDF]/PDF — name? integrable?

Suppose I have a random variable $X\in\mathbb R$ distributed according to a smooth nonzero probability density function (PDF) $f(x)$, with cumulative distribution function (CDF) $F(x):=\int_{\infty}^x ...
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Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
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267 views

Calculate expected value of CDF of a different Beta variable

Let $$ X_1 \sim Beta(\alpha_1,\beta_1) \\ X_2 \sim Beta(\alpha_2,\beta_2). $$ Let $F_X(x) = P( X \le x )$ be the CDF of $X$ and $\mathbb{E}_{X}(\cdot)$ be expectation with respect to $X$. How to ...
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179 views

Testing for Normality (CDF)

I was reading an article about using the CDF to calculate the area between 2 points on the normal curve. They gave a sample of 7 for illustration purposes: ...
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682 views

CDF of the ratio of two correlated $\chi^2$ random variables

It is well known that the sum of a series $m$ of squared standard independent normal random variables follows a $\chi^2$ dstribution with $m$ degrees of freedom. It is also true that the ratio of two ...
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794 views

Why it is better to use the cumulative distribution to compute distances?

In the comments of this question, it was pointed out that, when comparing two distributions, it is more natural and more general use the cumulative distribution (CDF) instead of the distribution (PDF)....
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What does weighted cumulative frequency distribution mean?

I have two sets of data (temperature and catch) and I am following a proposed method in an article I am reading on the empirical cumulative function (ECDF) analysis. Firstly, I have derived the ECDF ...
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Integral of difference of density functions of two Continuous Random Variables goes to 0

The problem says : Let $(X_n)_{n=1}^\infty$ be a sequence of continuous random variables with probability density functions $(f_n)_{n=1}^\infty$ , and let $X$ be another continuous random variable ...
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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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55 views

Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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91 views

Computing the CDF of the minimum of particular dependent random variables

For each $i=1,\dots,n$ let $Z_i\sim\text{Poisson}(\lambda_i)$, and suppose $\{Z_i\}$ are independent. Also for each $i=1,\dots,n$, let $\{Y_{ij}\}_{j\in\mathbb{N}}$ be an infinite sequence of iid ...
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Constructing a joint distribution from pairwise bivariate marginal distributions?

It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
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222 views

Why do we need to estimate the cumulative distribution function?

In statistic estimation we always works with densities. However, sometimes we need to estimate the cumulative distribution function? why? What is the benefits of estimating the cumulative distribution ...
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Finding PDF from CDF

I just got really unsure, can someone confirm/rectify? I have the CDF defined as $F(x)= \begin{cases}0, &\text{if}~x < 0,\\ 4x^2 &\text{if}~ 0 \leq x < \frac{1}{4} \\ 1-\frac{4}{3}(1-x)^...
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1answer
375 views

Confidence interval for the distance between two CDF

Let $X$ and $Y$ be two real random variables with cumulative distribution functions $F_X$ and $F_Y$, respectively. Define $d = \sup(F_X(x) - F_Y(x))$. If $\hat F_X$ and $\hat F_Y$ are empirical ...
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Cumulative distribution function for the product of two random variables

Given two random variables $x, y$, each with the probability distribution functions $p_x(x)$, $p_y(y)$, then if $z = xy$, then $p_z(z) = \int p_x(x)p_y(z/x)\frac{1}{|x|}dx $. Is there a similar proof ...
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Geometric construction of copula - question regarding C-volume

I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: http://www.stat.ubc.ca/...
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How to calculate CDF of g(X)

Let $X$ a random variable with distribution $F_X(x)$ $$Y=g(X) = \left\{ \begin{array}{lr} X-c & : X > c\\ 0 & : -c < X \le c \\ X+c & : X \le -c \end{array} \right\}$$ ...
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108 views

Bivariate CDF estimation

If I have a sample $(X_i,Y_i)_{i=1}^n$ I can estimate the joint CDF by: $$F_{X,Y}(x,y) = \frac{1}{n}\sum_{i=1}^nI[X_i\leq x,Y_i \leq y]$$ Assume now I observe only $(X_i)_{i=1}^n$, but I know that $Y\...
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864 views

CDF of conditional distribution

Suppose that $X$ and $Y$ are iid normally distributed and $a$ is a scalar. What is $\Pr(Y+aX<0 | X>0)$?
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456 views

Computational shortcuts/approximations for beta-binomial and beta-negative binomial CDFs

There are several simplifications that can be done so that computing cumulative distribution functions of beta-binomial and beta-negative binomial distributions, but still computing CDF as $F(x) = \...
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Can we sample from both pdf and cdf?

my question is quite generic. I am currently studying the algorithms calculating random numbers from distributions: In inverse transform method we get the cumulative distribution function in the end ...
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How can we evaluate the following CDF?

First let us consider the following regression \begin{equation} y_t=\beta'x_t+\varepsilon_t,\quad t=1,...,n \end{equation} where $x_t$ is a $k\times 1$ vector of "fixed" regressors and $\beta$ is ...
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109 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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129 views

Efficient sampling from a multivariate Gaussian Mixture distribution for a given CDF level

I have a multivariate Gaussian Mixture (GM) distribution. I am wondering if there is any more efficient way of drawing samples (i.e., identify the iso-surface) from a multivariate Gaussian Mixture ...
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Must the domain of a CDF be $\mathbb{R}$ or can it also be a strict subset?

So my question is whether the domain of a cumulative distribution function has to be $\mathbb{R}$ or whether it can also be a strict subset. The reason I'm asking is because I'm currently going ...
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Under what conditions does the two-sided DKW inequality become a strict equality?

If the two-sided DKW inequality is tight, then there should be a choice of distribution and sample size where the equality holds. What is it?
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What is the problem in my CDF derivation?

Let $Z = \frac{XY}{aX+bY+c}$ where the random variable $X$ and $Y$ follows gamma distribution such that $X\sim G(\lambda_x,\theta_x)$ and $Y\sim G(\lambda_y,\theta_y)$ The CDF of $Z$ can be ...
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195 views

Finding probability of a point using bivariate copula density

I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
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Joint distribution of multivariate normal

Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$. \begin{align*} P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\ &...
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How to add dependence between random vectors using a copula?

I understand that copulas can be used as a tool to add any conceivable dependence to a pair of random variables. However, I would like to add some dependence between two random vectors. Let us ...
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1answer
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It is true that $\mathbf X \sim F_X \Rightarrow F_X(\mathbf X) \sim U_{[0;1]}$; does the converse hold for multivariate $\mathbf X$?

For a univariate real-valued random variable I am pretty sure that the converse holds. Consider a multivariate $\mathbf X$ with values in $\mathbb R^n$ with measure $\mu(X)$ and its multivariate CDF ...
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Equation for Inverse Poisson CDF

I am attempting to calculate quantile probabilities. I.e., the value above which there is only a 1% chance occurrence for an arrival process. The R code is pretty straight forward with say a lambda = ...
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Distributions with some conditions

Suppose $F(x)$ and $G(x)$ are some CDF's with support $[0,1]$. I'm looking for relationship between $F$ and $G$ when it satisfies those following conditions: $$G^{-1}\bigg(\frac{1}{2}\bigg)\int^1_0F(x)...
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Conditions for second-degree stochastic dominance for uniform distributions

Currently I am reviewing first- and second-degree stochastic dominance. Suppose that we have two uniform distributions $$F^1(y) = \frac{y - a_1}{b_1 - a_1}$$ and $$F^2(y) = \frac{y - a_2}{b_2 - a_2}$$...
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Conditional cdf

I want to know that how conditioning will affect the CDF of dependent random variables. More specifically, let's suppose, $\Gamma_R={g\over A}$ and $\Gamma_D={g\cdot h\over B}$, where $g$ and $h$ are ...
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Joint CDF of random variables vis-a-vis that of their order statistics

Suppose $\{X_i\}_{i\in 1\ldots n}$ are $n$ independent, non-identically distributed RV's. Let $X_i \sim f_i(x) \mathbf{1}_{[0,1]}$, where $f_i$ is the $i$-th parent supported on $[0,1]$. I am ...
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What is the intuition behind this pmf/pdf abstraction?

When the writer wishes to show a sum/integration that applies similarly to both pmfs and pdfs, they write it like this: $\quad \int (whatever)\ dF_X(x) $ Such that this becomes either $\sum_x (...
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133 views

About deriving PDFs from CDFs

Suppose I have some continuous random variable $X$. Further, suppose I am interested about a transformed random variable $Y = g(X)$ where $g$ is some increasing function. If I know the CDF of $X$, I ...
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best visualization way to show a PDF+CDF to non-math people

EDIT: please ignore the actual formatting of the graph; it is meant as a demo and not "finalized". We are struggling to find the best way to present CDFs to our customers. They can read PDFs, but ...
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CDF on non-standardized t distribution

What is the cumulative distribution function of the non-standardized student's t distribution in terms of inverse scaling parameter? I have found a number of related equations online, but not this one ...
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Probability of Achieving a Count Level in Time Series Data

I have some time-series data that displays a count value for every day: These count values begin at 1 or -1 and will continue to count up (or down) if conditions in the time series are met. If the ...
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Kolmogorov–Smirnov “with perturbation”

Let $F$ be known continuous CDF of a continuous R.V. and $F_n$ represent the empirical CDF for sample of size $n$, hypothesized to be drawn from $F$. The Kolmogorov–Smirnov statistic is $D_n :=sup |...
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Inconsistency in output from my implementation of noncentral t CDF and R's pt()

I am trying to implement a noncentral t CDF as expressed by Guenther (1978), Lenth (1989), and the Wikipedia article on the non-central t in R. I have got my algorithm half working: when the signs of ...
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139 views

Asymptotic convergence involving EDF

Please help me proving this: Suppose that $Y_1,\ldots,Y_n$ are i.i.d. nonnegative RV's with CDF $F$ and $E(Y_i)=\mu<\infty$. Let $y_1,\ldots,y_n$ be a realization from which an EDF $\hat{F}$ is ...
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Expected Value of cumulative distribution function

Let $\varepsilon$ be a Gaussian distributed random variable with mean $\mu_0$ and standard deviation $\sigma_0$. Is it possible to compute/approximate the expected value $$ \begin{eqnarray} & &...