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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Approximate Metropolis algorithm - does it make sense?

Some time ago Xi'an asked What is the equivalent for cdfs of MCMC for pdfs? The naive answer would be to use "approximate" Metropolis algorithm in form Given $X^{(t)} = x^{(t)}$ 1. generate $Y ...
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Conditional CDF and PDF equivalence

Imagine there are two random variables $X$ and $Y$, with CDFs given by $F(x,y)$, and corresponding marginals $F(x)$ and $F(y)$. There exists a PDF, given by $f(x,y)$, with marginals $f(x)$ and $f(y)$. ...
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Comparing two CDF with normalized Kolmogorov-Smirnov

The Kolmogorov-Smirnov (KS) test is a well-known metric if you want to compare two CDF. It computes the maximum difference between the two. The main concern I have ...
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825 views

Why is the empirical cumulative distribution of 1:1000 a straight line?

Why does plot(ecdf(1:1000)) produce a straight line? Since Fn($x_n$) = $x_1$/(total sum) +$x_2$/(total sum) +...+$x_n$/total sum = ($x_1+x_2+x_3+...+x_n$)/total sum. the fact that Fn(200) roughly ...
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Is it better to look at ECDFs than PDFs when exploring empirical sample distributions?

I know that when plotting CDFs, ECDFs, or PDFs by using finite samples, we must be doing some form of interpolation. As far as I guess, empirical cumulative density functions (ECDFs) perform linear ...
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24 views

Multivariate CDF

I am having difficulties understandomg the following equality: $$\begin{align*} &P\Big(\frac{Z_{1}}{n}+\mu>z,Z_{2}>y,\frac{1}{n}\sum_{i=3}^{m}Z_{i}+\mu \leq w\Big) \\ &=\int_{n(z-\mu)}^...
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1answer
48 views

Calculating probability of displacement using two CDFs

My knowledge of stats is fairly basic, so you please bear with me! I'm trying to calculate the CDF for the vertical displacement of a (light, small) object floating in a wave tank. I have ...
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31 views

Bayesian Data Analysis: Section 3.4 Sampling from the joint posterior distribution Example

For reference: $$ p(\sigma^{2}|y) \propto \tau_n N(\mu_n | \mu_0, \tau_0^{2}) \text{Inv}-\chi^{2}(\nu_0, \sigma^{2}_0) \prod_{i=1}^{n} N(y_i|\mu_n,\sigma^{2}) \tag{3.14} $$ The book states: As ...
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1answer
33 views

Interpolate a CDF to get an interpolated hazard rate, or interpolate the hazard rate directly?

My problem is that I need to do an interpolation. Eventually, I will work on the hazard rate, but I do not know if it is better to interpolate the CDF or the hazard rate. Let me explain better. I've ...
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16 views

Random right cenoring

I observe a series of values from different trials ($Y_1, ... Y_N$). All values come from the same distribution ($F(\cdot)$). Trials do not have the same number of values ($N$ differs across trials)....
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Are CDFs more fundamental than PDFs?

My stat prof basically said, if given one of the following three, you can find the other two: Cumulative distribution function Moment Generating Function Probability Density Function But my ...
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Question regarding distributions and prediction

During my learning about distributions I came up with the following prediction algorithm. Given a dataset, for example a plot of stock prices to time. Select a range of values from the graph (for ...
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9 views

Create similar CDF utilizing less samples

My problem is as follows: I'm measuring the D-Stat between two CDFs (lets call them A and B), but I was thinking if it was possible to create a very similar CDF to the original A, utilizing less ...
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1answer
31 views

Truncated distribution and mass points

If you have some continuous distribution $F(x)$ and you truncated the distribution at some inclusive point $m$ i.e. $F(x | x\ge m)$ does the truncated distribution have a mass point at $m$? Also ...
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1answer
21 views

uniform histograms in copula approach

I would like to model some time series. For this purpose I have the marginal distribution and want to use a gaussian copula to build the dependency. Following a tutorial the following should give me ...
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63 views

Variance of a Cumulative Distribution Function of Normal Distribution

Suppose, $X\sim N(\mu,\sigma^2)$. Can anyone help in finding the following : $\text{Var } \bigg(\Phi\big(\frac{X + c}{d}\big) \bigg)$ ? Here, c and d are positive. Here, $\Phi(x)$ is the "...
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1answer
50 views

How to think about random CDF?

Chi (2014, "The Value of Information and Dispersion") defines a RV $X^*$ by the continuous distribution of (footnote 7, pg. 6): $$ G(x) = \begin{cases} (1-p) U_1 & \mbox{if } x=0 \\ 1-p & \...
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28 views

Which CDF to use? [closed]

Hey guys I'm preparing a Statistics assignment and I have a very large data set showing the ground temperature over the years. Data set consists of temperature and temparature uncertainty. Example is ...
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1answer
42 views

Understanding the CDF of the Exponential from the PDF?

I was trying to get the CDF of the exponential through the pdf. I know that the relationship between the pdf and the cdf is that the pdf is the derivative $ \lambda \exp(-\lambda x) $. But I don't ...
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1answer
57 views

Proving the probability integral transform without assuming that the CDF is strictly increasing

I know that the proof of the probability integral transform has been given multiple times on this site. However, the proofs I found use the hypothesis that the CDF $F_X(x)$ is strictly increasing (...
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16 views

Cumulative distribution discrepancy

The CDF of a normal variable is $P(X \leq x)$, where $X$ is a random variable. This also written as $\Phi (x)$ so if $\Phi (\cdot)$ is the normal CDF, then $\Phi (0)$ is $P(X<0) = 50 \% $ ...
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1answer
20 views

Prove that the cdf is strictly increasing on the support if the support is a finite interval

I am trying to prove that that for a distribution $P$ with cdf $F$, if the support of $P$ is a finite interval $I=[a,b]$ then $F$ will be strictly increasing on $I$. It's probably a simple ...
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1answer
28 views

solution to mixture CDF / inverse CDF of finite mixture

I am currently numerically solving the follwing equation, which is a convex combination/finite mixture of two marginal CDFs $F(x)$ and $G(x)$ (actually they are joint CDFs but all arguments but x are ...
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34 views

Get CDF from PDF

I am trying to evaluate the integral of pdf to get CDF but because of the absolute value, everytime I fail. can someone help me out ? thanks please note this is not HW, it is not practice problem ...
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19 views

How to compare binned CDFs?

We want to compare binned CDFs from sampled data. Each CDF is 101 numbers representing percentiles, i.e., $P(x<a_k)=k/100$ where $k=0..100$. The sample size is 1-3k (unknown at comparison time ...
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33 views

Obtain marginal CDF from joint CDF by simulation

How can I evaluate the marginal cumulative distribution function of a set of random variables for which I do not have the CDF in closed form. I can, however, simulate from a joint distribution ...
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46 views

Expanding Joint probability distribution function having dependent random variables

Let $Z_1$ and $Z_2$ denotes two dependent random variables defined as \begin{align} Z_1&=\frac{XY}{aX+bY+c}\\ Z_2&=\frac{XY}{uX+vY+w} \end{align} where $X$ and $Y$ are independent ...
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20 views

CDF of transformed random variable

Given two random variables $X, X'$ that are connected via a transformation $T: X \to X'$, I would like to know if the following is true: $$ \text{cdf}_{X'}(x') = \text{cdf}_X\left(T^{-1}(x')\right). $$...
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1answer
81 views

What is the link (discrepancy?) between these PDF/CDF and p-value distributions?

I have created a mixed distribution model comprising 80% $H_0$ plus 20% $H_1$ to illustrate the link between the expected proportions of true and false positives and negatives in the PDF, CDF and p-...
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2answers
38 views

Generating data from a Beta-Binomial distribution by inverting CDF in R [closed]

I am trying to generate data from a beta-binomial distribution by inverting its cdf in R. The code I have written to calculate the cdf seems to be working fine for most cases, but gives me values ...
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1answer
37 views

Variance from cdf

I have an empirical cumulative probability distribution function $F$ for a random variable $X$ (non-negative). Is it possible to estimate the variance from $F$ ? I already estimated, numerically, the ...
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1answer
48 views

Compute cumulative distribution using triangular kernel function

Suppose that we use the kernel density estimate ${\hat{f}}_h\left(x\right)=\frac{1}{nh}\sum_{i=1}^nK\left(\frac{x-X_i}{h}\right)$ With the kernel function $K\left(u\right)=\left(1-\left\vert{}u\...
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2answers
62 views

How to find the CDF of a random variable uniformly distributed around another random variable?

I'm working on some game theory models of incomplete information (which I've posted about a few times here). I think this question is pretty straightforward though, so the actual context is ...
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1answer
94 views

How to compute the CDF of this random variable?

I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from ...
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1answer
20 views

bivariate normal distribution & change of units

I need to find the bivariate normal CDF at the point (1 cm, 1 cm), i.e. the half-open interval $[-\infty,1] \times [-\infty,1] $ , where cm = centimeter, with mean = 0, covariance = Id. mvncdf( x = ...
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1answer
39 views

What is the density of the $m$'th element of a sorted vector of $n$ uniformly distributed random variables

$X_1, X_2, ..., X_n$ are independent and uniformly distributed on $[0, 1]$. Sorting them yields a vector, whose first and last element have densities that are just the derivatives of products of CDFs. ...
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49 views

Densities and Cumulative Distribution Functions

Has anyone seen this notation before? What does it mean? $\int_{0}^{\infty} f(x) G(x) dx$ $f(x)$ is a density and $G(x)$ is a cumulative distribution function
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1answer
49 views

Finding the expected value of the cdf?

I have this question on a homework assignment, and I'm not sure how to solve it. Assume $X \sim \exp(\lambda=2)$, define $Y=F(x)$, where F(x) is the cdf function of $X$. Calculate the expected value ...
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64 views

P values and calculating residuals in chi-squared contingency tables

Given an n x m contingency table I calculate a chi-squared value and probability for that table telling me if there are cells that differ significantly from the mean. I go to calculate the adjusted ...
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1answer
26 views

Directly obtaining marginal cdf directly from joint cdf

I know how to obtain marginal PDF $f(x)$ from $f(x,y)$. Just integrate over $y$. But is there a way to directly obtain marginal CDF $F(x)$ from $F(x,y)$? Do I need to calculate marginal PDF $f(x)$ ...
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1answer
38 views

Uniquely determine the two parameters of a distribution given pre-determined probabilities for two disjoint subsets of its support

Let $F(x| \alpha, \beta)$ denote the cumulative density function of a probability distribution. Let $[a,b]$ and $[c,d]$ be two disjoint subsets of the support of $F$. Suppose that $F(b) - F(a) = p$ ...
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1answer
56 views

Conceptual question on random variables with the same cdf

Given a continuous random variable $X$ let $Z = F_X(X)$, where $F_X(s) = P(X \leq s)$. Then $P(Z \leq z) = z$ and we say that $Z$ is uniformly distributed. Say we have another continuous random ...
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71 views

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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72 views

minimum cdf given mean and variance

Suppose I have some distribution with mean $\mu$ and variance $\sigma^2$. How to prove that for the distribution to have min $cdf$ over all the distributions with mean $\mu$ and variance $\sigma^2$, ...
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78 views

How do I determine parameters of normal & lognormal distribution given two points?

Assume I have two values which represent two quantiles for the same lognormal and/or normal distribution. How can I determine the parameters of the distribution? ...
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1answer
156 views

How are the Error Function and Standard Normal distribution function related?

If the Standard Normal PDF is $$f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$ and the CDF is $$F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2}\mathrm{d}x\,,$$ how does this turn into an error ...
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Is it reasonable to estimate the CDF of a random variable with the help of kernel density estimation?

Given a set $S$ of samples from a random variable $X$ with pdf $f$ and cdf $F$, is it reasonable to estimate $F(x)$ by finding an estimation $\hat{f}$ of $f$ via kernel density estimation on $S$, and ...
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2answers
39 views

Numerically finding confidence interval bounds

I am asking an R question here on the basis that statistical expertise is needed, i.e., If the language is statistically oriented (such as R, SAS, Stata, SPSS, etc.), then decide based on the ...
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313 views

Probability of failure

A structure will fail if subjected to a load greater then its own resistance: failure := load > resistance We can assume that the load and the resistance are ...
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73 views

(CDF Quantile) Understanding an example from a swirl lesson

I need to understand an explanation from a swirl() lesson in course "Statistical Inference", menu option "Probability2". At 76% of completion, there is a question: "The quantile v of a CDF is the ...