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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Two sample one-sided Kuiper Test and KS-statistic

With the KS-Test it is possible to conduct a two sample one-sided test between two different random samples $A$ and $B$ to test whether one CDF is larger or smaller than the other, i.e. is $CDF_A$ ...
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Why is the CDF of a sample uniformly distributed

I read here that given a sample $ X_1,X_2,...,X_n $ from a continuous distribution with cdf $ F_X $, the sample corresponding to $ U_i = F_X(X_i) $ follows a standard uniform distribution. I have ...
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Difference between density and probability [duplicate]

What is the difference between the density and probability? I have tried R in which I can use both pnorm and dnorm for the ...
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How To Calculate and Print CDF and Lower Bound CDF value for some t (time) with Minitab (without examining the plot)

I'm a reliability engineer. Cumulative Distribution Function (CDF) in statistics stands for the unreliability (probability of failure at a time value) of the item being tested. Unreliability is very ...
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Cumulative distribution functions (cdfs) range uniformly [duplicate]

I am confused .. how does this happen? "continuous cumulative distribution functions (cdfs) range uniformly over the open interval (0,1).". How does the cdf range "uniformly" (each value having the ...
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66 views

Is every cumulative probability density function Borel measurable?

I have seemingly simple question, which does not need to have a simple answer :) Is every cumulative probability density function Borel measurable?
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23 views

Translate exponential distribution into normal distribution [closed]

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
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1answer
47 views

How to compute $P(|X - E_Y[h(y)]| < c)$?

Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and ...
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Tail of the inverse cdf

I am almost sure I have already seen the following result in statistics but I can't remember where. If $X$ is a positive random variable and $E(X)<\infty$ then $\epsilon F^{-1}(1-\epsilon) \to 0$ ...
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3answers
86 views

How to derive the cdf of a lognormal distribution from its pdf

I'm trying to understand how to derive the cumulative distribution function for a lognormal distribution from its probability density function. I know that the pdf is: $$f(x) ...
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47 views

Identify outliers with median-absolute-deviation for timeseries data

I am having trouble understanding this particular method of detecting outliers in a time series. Below is the problem: I have a region-of-interest containing 15 voxels. Each voxel contains values ...
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66 views

Finding the cumulative distribution of a mixture distribution of discrete and continuous variables

If I have a random variable that with probability 1/3 is a $U(1,2)$, with probability $1/3$ is $U(2,4)$ and with probability $1/3$ is a discrete rv that takes value 2 with probability $0.4$ and 3 with ...
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Pointwise convergence of the cdf of normal random variables

For a sequence $X_1, X_2, \dots $, Let $F_n(x)$ denote the cdf of $X_n$. Suppose our sequence is $X_n \sim N(0,n) $ then for all $x$ the point-wise limit of $F_n(x)$ is $\frac{1}{2}$. How would one ...
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66 views

limit involving CDF and Quantile functions

Suppose $F(z,\Theta)$ is a continuosly differentiable cumulative density function where $z$ is a random variable defined over $[0,\infty)$ and $\Theta$ is a parameter of the distribution. I have a ...
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1answer
160 views

Deriving Density Function (pdf) from Distribution Function (cdf)

A random variable $V$ has the distribution function: $$ F(v) = \begin{cases} 0, & \text{for $v<0$ } \\ 1-(1-v)^A, & \text{for $0\le v\le1$ } \\ 1,& \text{for $v>1$ } \\ \end{cases} ...
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Inverse CDF of a Student's t-Distribution

I'm trying to implement a mathematics library in C# .NET which includes some basic statistics. I need to be able to determine the inverse CDF of a Student's t-distribution from a given ...
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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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1answer
59 views

Maximum likelihood of constrained distribution

A random variable $X$ is represented by the following CDF: $F(x)=(1+\frac{1}{x^2})^{-\beta}$ , $x\in(0, \infty), \beta>0$ Question: How do you get the MLE of $P(X>1)$ for the distribution? I ...
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Goodness of Fit Using $CDF$

I don't have a tremendous amount of experience with testing goodness of fit, and I've just begun studying it rigorously. A though occurred to me, and since it occurred to me, I'm sure it exists, and ...
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103 views

Simulate from a truncated mixture normal distribution

I want to simulate a sample from a mixture normal distribution such that $$p\times\mathcal{N}(\mu_1,\sigma_1^2) + (1-p)\times\mathcal{N}(\mu_2,\sigma_2^2) $$ is restricted to the interval $[0,1]$ ...
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1answer
24 views

How to compute the $\chi^2$-table values?

How one can compute the values of $\chi^2$-tables? I saw two tables where was given for example that if degree of freedom is $1$ and $p=0.001$ then table value is in one table $10.827$ but in the ...
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1answer
122 views

How to compute joint cdf of an empirical copula? (Updated with more info)

lets suppose a bivariate empirical copula as: for a set of data of example data we can plot it like this: How can we compute the joint cdf of this empirical copula which should like this: Thank ...
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1answer
265 views

Is dunn.test a suitable alternative to kruskalmc in pgirmess package?

I'm trying to run the krushkalmc method after running kruskal.test as part of my analysis with the Kruskal-Wallis rank sum test. I have data with a small sample size that also does not have a normal ...
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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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1answer
70 views

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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70 views

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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166 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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1answer
101 views

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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71 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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1answer
368 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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1answer
58 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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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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45 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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1answer
225 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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1answer
24 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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1answer
42 views

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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77 views

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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122 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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71 views

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 ...