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 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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35 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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43 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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22 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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18 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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33 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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31 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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68 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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63 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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54 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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47 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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27 views

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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33 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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254 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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34 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 ...
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80 views

Weibull distribution in R

I wanted to know how do I calculate exact values for Weibull distribution in R. For example, say $X$ ~ Weibull(3,5), and I want to calculate $P(X\gt 7)$. How do I do that? I want to estimate $P(X\gt ...
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13 views

How to compute the VaR(a) of the following cdf

I have to compute the mean value, the variance, the VaR and ES of the CDF F(x)= 1- (3/2x)^(-3) if x>= 3 F(x)=0 if x< 3 The mean and variance I have computed, but what about computing the value at ...
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160 views

limit of $x \left[1-F(x) \right]$ as $x \to \infty$

I am wondering about showing that the limit: $$ \lim_{x \to \infty} x\overline{F}(x) =0 $$ where $\overline{F} =1-F$ is the tail distribution function, $\overline{F}(x)=1−F(x)$, where $F$ is the ...
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36 views

showing that a function is a copula

In general, is there an easier way of showing that a function is a copula than showing that: $C(u_1,\dots,u_d) =P(U_1\le u_1,\dots,U_d \le u_d) \quad$is nondecreasing in each $u_i \in [0,1] $ ...
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36 views

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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125 views

Notation for CDF of the binomial distribution

The CDF for $X \sim \operatorname{Bin}(n,p)$ is $I_{1-p}(n-k,1+k)$. What does the $I$ mean?
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279 views

A possible mistake in a conditional probability derivation

The following is a derivation of a density from a paper I am currently studying. Sorry for the bad quality, it is quite an old paper. I need to clarify that $R$ has the standard exponential density in ...
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43 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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Kolmogorov-Smirnov vs simple area difference estimation to compare two statistical models

Could anyone explain, or point some references that I can locate this, how KS compares to the relative area estimation method? It is known that Kolmogorov-Smirnov, among other famous similar ...
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116 views

Empirical verification of the probability integral transform

I just discovered when working on Copulae that it was common knowledge that if $X$ is a continuous random variable with probability density function $F_{X}$, then $Y=F_{X}(x)$ follows a uniform ...
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30 views

Weibull PDF and CDF value evaluated at time t? [duplicate]

Given the Weibull Distribution with PDF and CDF defined as follows: \begin{gather*} f(t) = \frac{\alpha }{\beta }\left(\frac{t}{\beta }\right)^{\alpha -1}e^{-\left(\frac{t}{\beta }\right)^{\alpha }} ...
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20 views

survival function in terms of copula

Is the survival copula the equivalent of the multivariate survival function? In other words, can I write $\bar{C}(u,v) = S(u,v) = u+v-1+C(1-u,1-v) $ where $\bar{C}$ is the survival copula, $S$ the ...
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40 views

How do I estimate a smooth cdf from a set of observations?

I have a set of observation, let's call it $X$ and would like to fit a cdf to it. $X$ has a distribution which is roughly approximable with the normal distribution. This CDF should correspond to a ...
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65 views

Let $X$ be a random variable and $f$ an invertible function. Then the CDF of a random variable $Y=f(X)$ always exists?

Let $X$ be a random variable and $f$ and invertible function. The cumulative distribution (CDF) of $X$ is defined as $$F_{X}(x) = \mathrm{P}(X\leq x).$$ The CDF of $Y=f(X)$ is then $$F_{Y}(y) = ...
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130 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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58 views

Compute the CCDF of $p(x)=\frac{1}{m} \sum_{k=1}^{m-1} \frac{k!}{(k+n-m)!} [L_k^{n-m}(x)]^2 x^{n-m} e^{-x}$

I have the following PDF: \begin{align} p(x)=\frac{1}{m} \sum_{k=1}^{m-1} \frac{k!}{(k+n-m)!} [L_k^{n-m}(x)]^2 x^{n-m} e^{-x}, \quad \quad x \ge 0, \end{align} where $L_k^{n-m}(x)$ is the associated ...
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Follow up questions on the Distribution of a ratio of uniforms

I have a follow up question regarding this question (Distribution of a ratio of uniforms: What is wrong?) For the first method, and in case of z > 1 , How did he set up the limits of the integration? ...
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relationship between rapidly varying tails and relatively stable distributions

Suppose a random variable X has cdf $F$ has rapidly varying tail $\overline{F} =1-F$, such that: $$ \lim_{x \to \infty} \frac{\overline{F}(x\lambda)}{\overline{F}(x)}= 0 $$ if $\lambda >1$, and ...
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24 views

code for first and second order stochastic dominance

I trying to test if one variable stochastically dominates the other both first and second order, however, I don't have the code to perform the test and I am not very familair with programming language ...
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206 views

How to fit a function to a CDF in R?

I've been given a dataframe that contains data for a CDF. The column X contains the 250 $X$ values, and the column P contains ...
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91 views

How are percentiles distributed?

I was taking a look at this page, and I can't seem to understand why the frequency plot of the percentiles is uniformly distributed. Distances between percentiles are not equal, so why is the ...
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85 views

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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177 views

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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244 views

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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27 views

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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84 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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58 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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50 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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107 views

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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136 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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202 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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191 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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53 views

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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79 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 ...