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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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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59 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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131 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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56 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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54 views

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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58 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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19 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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71 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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69 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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52 views

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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36 views

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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42 views

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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51 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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39 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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21 views

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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158 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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281 views

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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62 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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51 views

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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56 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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228 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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43 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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46 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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43 views

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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43 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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213 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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87 views

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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40 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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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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95 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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48 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 ...
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132 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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55 views

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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13 views

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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253 views

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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70 views

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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27 views

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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26 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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244 views

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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67 views

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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90 views

Estimating the area between two ecdfs

I wish to calculate the area between two ecdfs in R (see below): ...