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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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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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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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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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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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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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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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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41 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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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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71 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 ...
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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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34 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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47 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 ...
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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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92 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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15 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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35 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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45 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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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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38 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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21 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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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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37 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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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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68 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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69 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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82 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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37 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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284 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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(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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101 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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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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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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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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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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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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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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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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174 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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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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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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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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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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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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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 ...