CDF is an acronym for 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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What is the joint CDF of a Vine copula?

Consider a 4 dimensional D-Vine Copula with the following density function: what will be the joint cdf function? The pdf function is from Aas et al., 2006 ...
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CDF of the function of a random variable

I haven't been able to find useful information on this. I was just wondering what would be the distribution of a function of a random variable. For example, what would be the distribution of the ...
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Probability Interval for F(x) with Parameter Estimates from Bayesian Analyses

Problem: I estimated the shape $\alpha$ and scale $\lambda$ parameter of the Weibull distribution using Bayesian methods. That gave me a marginal posterior distributions for both parameters. ...
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Does the position at which maximum distance occurs in a KS test make a difference?

From my understanding of the KS test, fromt the CDF of two datasets, it measures the distance between the two distributions at various points and and compares the 'maximum distance' to a predefined ...
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When to use CRPS (Continuous Rank Probability Score)? What are the alternatives? What are the advantages and disadvantages?

Please correct me if I'm wrong, crps is new for me. I want to understand it better. We have to minimize crps, which is based on the cumulative distribution function of the data. While the information ...
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On Kolmogorov's Theorem In Time series theory and methods (1990)

I am following Time series theory and methods, Brokwell and Davis (1990). And theorem 1.2.1 called by the text Kolmogorov's Theorem is only stated but not proven. I will rewrite it here: The ...
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Results stronger than Dvoretzky–Kiefer–Wolfowitz inequality?

There is the DKW inequality which controls the extent to which the empirical cdf of a sample from a real-valued random variable differs from the true cdf. Are there any stronger results (i.e. ...
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149 views

What is the limiting distribution of exponential variates modulo 1?

I have tried to find the limiting distribution of $X_n\sim\text{Exponential}(\lambda/n)$ by finding the cdf and taking the limit. I got: \begin{align*} F_{X_n}(X)) = \int_{0}^{X} \frac{\lambda}{n} ...
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Why does a Cumulative Distribution Function (CDF) uniquely define a distribution?

I have always been told a CDF is unique however a PDF/PMF is not unique, why is that ? Can you give an example where a PDF/PMF is not unique ?
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How to obtain the quantile function when an analytical form of the distribution is not known

The problem comes from page 377-379 of this [0] paper. Given a continuous distribution $F$ and a fixed $z\in\mathbb{R}$, consider: $$L_z(t)=P_F(|z-Z|\leq t)$$ and ...
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Get probability distribution function from density function

For a given density function, how does one find its distribution function? For example, I have a density function: $f(x)= \begin{cases} t ^2 / 9 & \text{if } t \in (0,3)\\ 0 ...
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Finding a Random Variable $X$ on the sample space with given cdf

I will state the problem first, then follow through with my work. Q: Suppose that the sample space is given by $S={w_1,w_2,w_3}$ where $w_1,w_2,w_3$ are three states of tomorrow's weather. We have ...
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48 views

CDF for list of numbers

I hope that someone can help me with this! I have a list of values as below: 82.1134 84.5516 91.1851 65.6035 69.971 92.4706 79.1505 93.0844 92.9598 and I need to ...
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193 views

Advantages of Box-muller over inverse CDF method for simulating Normal distribution?

In order to simulate a normal distribution, from a set of uniform variables, there are several techniques: The box muller; in which one samples two independent uniform distributions $(0,1]$ and ...
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36 views

How to Map Desired Confidence Interval to a Quantile value

I want to calculate the N% confidence interval for some time-series data set. I have the standard errors for this data series and the error variance of the time-series is assumed to be Gaussian. I ...
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32 views

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: ...
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What means “CDF of one group does not cross the CDF of the other” from the dunn.test description? [duplicate]

From the dunn.test manual: "CDF of one group does not cross the CDF of the other". What this means? I am not a statistician, but I want to understand. What I found ...
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42 views

Exclude Some samples for calculating CDF

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
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Are there non-trivial settings where the MAD statistic has a closed-form density?

The MAD statistic of an iid sample $(x_1,\ldots,x_n)$ is defined as the median of the absolute deviation from the median: $$ ...
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209 views

Area within a given number of standard deviations from given mean

I have a variable with mean value of 18.85 and standard deviation of 1.45. I want to define the area that is covered by 1.45 standard deviations left and 1.45 standard deviations on the right side ...
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Verification of an optimal parameter from an empirical CDF

Suppose we have the following model for the variable $V_5$: $$V_5 = \prod_{k=1}^5(e^{\mu + 0.2X_k}+0.05e^{0.05Y_i - 0.00125}), X_i,Y_i\sim N(0,1)$$ What I wish to do is to solve the problem ...
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Fitting parametric CDF to ecdf

There is a random variable $X$, but the only data I observed is actually its empirical distribution function (at a suitably fine grid). That is, I only observe $\hat{F}(x)$:=$\#\{x\leq u\}\over N $ ...
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23 views

Joint PDF of a set of equations

I am looking for a way to find the joint pdf of vector $Z=[Z_1,Z_2,Z_3,Z_4]$ where $Z_1= a_1 X_1^2 + a_2X_1Y_1+ a_3 X_1Y_2 + a_4Y_1^2 + a_5Y_2^2$ $Z_2= b_1 X_1^2 + b_2X_1Y_1+ b_3 X_1Y_2 + b_4Y_1^2 + ...
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PDF of sum of independent Gaussian variables

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
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How to make non-parametric distribution estimation with known, limited number of points of the CDF?

Is there any method to make non-parametric estimation of a cumulative distribution function (CDF) which actual points (not a sample) can be calculated numerically? I have a numerical method which can ...
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89 views

Convolution of random vectors

Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where $\mathbf{y}=A \ast B$, here $\ast$ ...
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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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126 views

Find the parameter of a Poisson, given the distribution function at a known value

Assuming a Poisson distribution, the probability ($\alpha$) that the result will fall within the range $0\ldots k$ is given by the following expression: \begin{equation} \alpha = ...
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Quad precision normal cdf and quantile functions

I'm looking to run the normal distribution cumulative distribution function and quantile function (its inverse) using the quadruple precision floating point format. Does anyone know of a library that ...
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How to determine CDF of Poisson function when only arrival time and service time are known?

If I can sample any given poisson arrival and service rate at any point and can sample any no of times. How just out of that sampling can I determine the CDF.
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Goodness of fit (cdf: empirical vs theoretical)?

I have a data-set with n = 90, probably follows the gamma distribution (and others). I used the maximum-likelihood estimation (MLE) to estimated the alpha and beta parameters of the gamma distribution ...
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CDFs for Right-Skewed Distributions

How does one determine the percentage of a sample less than or equal to some x value for a set of discrete data that appear to be right-skewed? For example, I have a number of data points, and if I ...
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What do I obtain if I subtract two CDFs?

Code: ...
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Percentage Beyond a Given Value for Empirically Defined Distribution

It is my understanding that standard deviation does not work well as a measurement for distributions that are heavily skewed. If I have a heavily right-skewed distribution, should I simply use the ...
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Difference between two vectors of unequal size

I have a predictive model function mPred and two different input variables matrices of the same size (nxm), iM1 and ...
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25 views

Dependent chi-squares vector: how to calculate cdf of $X_{(n)}$?

Consider a vector of central&1-degree Chi square distributed variable $(X_1, X_2,...,X_n)$, it is simple to calculate the cdf of $X_{(n)}$ (maximum of order statistics), when they are independent. ...
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convert lognormal Cumulative Density Function P90 and P10 values to mean and sigma [duplicate]

Practioners are used to defining lognormal distributions in terms of P90 and P10 cumulative density function values. To utilize these esperts' input I need to be able to convert these P90/P10 values ...
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Evaluate goodness-of-fit of estimation of Pareto-like distribution

I would like to evaluate the goodness-of-fit of the following (Pareto-like) distribution: $$ f(r) = \sigma \centerdot r^{-\rho} $$ The function estimates the population of cities given the rank $r$ in ...
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Estimating the area between two ecdfs

I wish to calculate the area between two ecdfs in R (see below): ...
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How to find pdf of a joint distribution in R?

$F(x,y) =\frac{1}{6}(x^2\, y+x\, y^2)\,,\quad 0\leq x\leq 2,\, 0\leq y\leq 1$ Above is the joint distribution given, how to find out cumulative distribution function of y? how to obtain joint ...
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Cumulative distribution function: what does $t$ in $\int\exp(-t^2)dt$ stand for?

I'm trying to teach myself how to quickly translate many different types of equations into VB, T-SQL and MDX code. Since I'm trying to build a skill, not just solve a single isolated problem, I'm try ...
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58 views

From joint cdf to joint pdf

We can get the joint pdf by differentiating the joint cdf, $\Pr(X\le x, Y\le y)$ with respect to x and y. However, sometimes it's easier to find $\Pr(X\ge x, Y\ge y)$. Notice that taking the ...
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51 views

Numerical approximation of percentiles from arbitrary pdf

Given an easily-computable probability density function $f(x)$, what algorithm can we use to numerically approximate percentiles? For instance, we might be looking for $x$ such that given $X \sim ...
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Convergence Issues for Bootstrap Distributions

the following is part of a proof from van der Vaarts book on asymptotic statistics: I want to show that if for a continuous distribution function F ...
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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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Area below probabilities

Let $p$ be probabilities and $D$ is the real How can I proof that the areas $$\int p \; d F_{p}(p|D=1) = \int (1-p) \; d F_{1-p}(1-p|D=1)$$ are equal. Where $F_{p}$ is the empirical distribution ...
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How can I calculate a discrete Cumulative Distribution MultiDimensional Array from a discrete Probability Mass Array when dimensions > 2?

I would appreciate any help in trying to calculate the Cumulative Distribution Array of a Probability Mass Array when dimensions > 2, essentially a discrete joint cumulative distribution from a sample ...
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CDF/ ECDF plot for data with two attributes

I have a data in the following format: $$ \begin{array}{rr} \textbf{colm_1} & \textbf{colm_2}\\ 3 & 1\\ 10 & 0\\ 3 & 0\\ 100 & 1\\ . & .\\ . & . \end{array} $$ colm_1 are ...
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What is the two-sample CDF of of $D^{+}$ and $D^{-}$ from the one-sided Kolmogorov-Smirnov Test?

I am trying to understand how to obtain $p$-values for the one-sided Kolmogorov-Smirnov test, and am struggling to find CDFs for $D^{+}_{n_{1},n_{2}}$ and $D^{-}_{n_{1},n_{2}}$ in the two-sample case. ...
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315 views

Discrete analog of CDF: “cumulative mass function”?

We call the integral of a probability density function (PDF) a cumulative distribution function (CDF). But what's the cumulative sum of a probability mass function (PMF) called? I've never heard the ...