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Questions tagged [inverse-cdf]

Inverse cumulative distribution function, known also as quantile function, for a given probability returns the value at which the probability of observing some outcome from the random variable is less than or equal to the given probability.

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Normal Quantile Function With a lower bound not equal to infinity

I was recently at a statistics competition and a question came up as follows: They drew a normal distribution with $\mu=7$ and the area between the values $7.75$ and $8.25$ equal to $0.12$. No other ...
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VaR/inverse cdf of transformation of normal variables

I have the following exercise to solve as good preparation for an exam: NOTE: $VaR_p(X)$ = Value at risk = $F^{-1}_X(p)$ Consider the bivariate normal random vector $(X_1, X_2)$. The marginals are ...
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Quantile Function

I have seen the definition of quantile function here, which is as follows (slightly modified): Let $X$ be a real-valued non-degenerate random variable with distribution function $F_X(x)=\mathbb{P}({X\...
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76 views

How to generate a Weibull distribution with inverse transform

Given $X\sim \text{Weibull}(\lambda,k)$, generate samples from the Weibull distribution using the inverse transform. We know $F_X(x) = 1-\text{e}^{-(x/\lambda)^k}$ for $x\ge 0$ with $\lambda,k > 0$...
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How to find the Inverse Transform of the Gumbel distribution

How does one find the Inverse Tranform of the Gumbel distribution? Let $X\sim \text{Gumbel}(\mu,\beta)$ with scale parameter $\beta>0$. The CDF is then $F_X(x)=\text{e}^{-\text{e}^{-(x-\mu)/\beta}...
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Expected value of joint quantile functions

Consider a population of individuals (not a sample). We are interested in two variables, $X$ and $Y$, which are independent. $X$ distributes with pdf $f(x)$ and CDF $F(x)$, and $Y$ distributes with ...
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Integral of the inverse of a CDF is the mean of the PDF? [duplicate]

While doing some calculations, I got this interesting result. Consider the CDF of the log-normal: $$ y(x) = \dfrac{1}{2}\left[1+erf\left(\dfrac{\ln\left(x\right)-\mu}{\sigma\sqrt{2}}\right)\right] $$ ...
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Confidence distribution for $\mu$ [closed]

Suppose that we have sample $x_{i}\sim N(\mu,\sigma^{2})\quad i=1,...,n$ and when $\sigma^{2}$ is known. A confidence distribution (CD) for $\mu$ can be written as follows (see, https://en.wikipedia....
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72 views

Is there any relationship between random numbers sampled from some continuous distribution D and the quantile function of D?

I've noticed that when I sample n numbers from a continuous distribution D, sort them, and plot against the quantile function of D, both curves seem very similar. I've tried this for a few ...
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268 views

How to calculate the inverse cdf for a sample with unknown distribution?

I'm stuck with an exercise that I'll just write down first: Based on a sample $x_1,...,x_{47}$, that can be considered coming from an unknown distribution, we study a qq-plot where the empirical ...
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1answer
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Using quantiles to estimate the parameters of a distribution: adjusting for unobserved extreme values

I with to estimate the parameters of a specified semi-infinite distributional family based on a sample drawn from that distribution. It is plausible that my sample median converges to the population ...
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How to generate data that have given conditional mean and conditional quantile using R?

Suppose I want to generate independent data $(y_{i},x_{i})$ such that the conditional mean of $y_{i}$ given $x_{i}$ is a quadratic function in $x_{i}$ and the $.25$ conditional quantile of $y_{i}$ ...
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Formula for Inverse T-distribution

I am trying to formulate an expression to calculate the critical value of a T-distribution for a given degrees of freedom. I have done so already for the Normal Distribution by considering the TI-84 ...
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62 views

Why do I get 'Inf' value while doing quantile mapping in MATLAB?

I am doing bias correction of rainfall data simulated by Global Climate Model. Since, rainfall data generally follows gamma distribution, I am estimating parameter ...
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305 views

Generate random numbers from a power-law/exponential distribution

I'm working through the paper Power-law distributions in empirical data by Clauset, Newman and Shalizi. On page 12, they generate random numbers from the following distribution: \begin{equation*} p(...
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1answer
72 views

How to simplify the derivation of the inverse cdf yielded from l'Hospital rule?

I am currently dealing with a proof of Pauline Barrieu`s Paper "Assessing Financial Model Risk" (page 19). At one point she applys the l'Hospital rule on a limit equation. We have some cumulative ...
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Nonparametric Quantile Regression for AR(1)-ARCH(1) process

I would like to estimate the conditional scale function $(\sigma_\tau(X_t))$ in a QAR-QARCH model represented by: \begin{equation} Y_t = \mu_\tau(X_t) + \sigma_\tau(X_t)\epsilon_t,\, t = 1,2,\ldots \...
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1answer
647 views

Monte Carlo Estimator for a quantile

I am trying to understand how to compute a crude Monte Carlo estimator for an $\alpha-$quantile. I have read the algorithm from the book Monte Carlo Methods and Models in Finance and Insurance, (Korn,...
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985 views

Generating Data from Arbitrary Distribution

If we want to generate a random sample according to a distribution $F$, we can generate a uniform random number on $(0,1)$ and invert it by $F$. This is due to the fact that, if $U$ is uniform on $(0,...
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Population and income quantiles and their relationship

This is not a statistical question, narrowly defined, but it is a terminological question concerning a set of topics that have attracted a great deal of statistical attention, and on which I am ...
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Equation for Inverse Poisson CDF

I am attempting to calculate quantile probabilities. I.e., the value above which there is only a 1% chance occurrence for an arrival process. The R code is pretty straight forward with say a lambda = ...
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Beta Binomial Inverse CDF

There are p groups of size $n_1, n_2, ... , n_p$ each with number of successes $x_1, x_2, ... x_p$ and number of failures $n_1 - x_1, n_2 - x_2, ... , n_p - x_p$. $X_i$ ~ $Binom( n_i, p_i)$, where $...
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Generate two distributions of data with consistent differences in quantile values

How can one generate two sets of data which are different at each quantile values (say decile)? Ex: 10th & 80th percentile is 4 & 22 respectively for the first set and the same should be 8 &...
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1answer
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What is the formula to calculate the nth percentile of Johnson-S$_U$ distribution with known parameters

I was wondering if there is a closed form to calculate the nth percentile of Johnson-S$_U$ distribution with known parameters? I know R could do this easily, but I want a formula instead so I can use ...
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How should I inverse of the Normal loss function L(z) to be able to implement this in software?

I have a question related to this question: How to inverse loss function L(z) to normal standard distribution (z) The question wasn't answered since it wasn't clear what was asked. However I know ...
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1answer
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Inverse Transformation Sampling with Gaussian

For inverse transform sampling, if you know the CDF of a probability distribution ($f_X$) that you want to sample, you can generate a uniform realization ($U$) from [0,1], and then according to the ...
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396 views

Input to fit a power-law to degree distribution of a network

I would like to use R to test whether the degree distribution of a network behave like a power-law with scale-free property. Nonetheless, I've read diferent people ...
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525 views

Copula Behaviour : Gaussian vs Student T -Numerical Stability

I would like to get your opinion on the following topic: I am comparing the behaviour of Gaussian and Student-t Copulas. I employ the follwing procedure: Simulate N=100,000 samples from a Student ...
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1answer
2k views

Generating a sample using inverse CDF method in python [closed]

How can I generate 1000-element sample using inverse CDF method? It should be easy but I'm a complete beginner. I know what it is mathematically but I don't know how to implement it in Python
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1answer
741 views

How to convert a Johnson normalized variable back to a marginal variable

Edit after @eric_kernfeld answer. I'd like to do Generate a time-series, for example, from a uniform distribution. Transform of non-normal variable to standard normal distribution. Fit an arima ...
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1answer
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What is inverse CDF Normal Distribution Formula

Do anyone know what is Inverse Cumulative Distribution Function of Normal Distribution formula or equation? I am looking at Google but did not find any good answer.
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What does the inverse Function in R calculate? [closed]

library(GoFKernel) x<-seq(0,1000,33) f <- function(x) pbeta(x, shape1=2, shape2=3) f.inv <- inverse(f,lower=0,upper=1) f.inv(.5) [1] 0.3857168 ...
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CDF of a Tweedie Distribution

The R package tweedie provides a way to estiamate the density and CDF of Tweedie distribution (using dtweedie and ptweedie, ...
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What are the advantages of an exponential random generator using the method of Ahrens and Dieter (1972) rather than by inverse transform?

My question is inspired by R's built-in exponential random number generator, the function rexp(). When trying to generate exponentially distributed random ...
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1answer
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Does q-function remain the inverse of CDF for any type of distribution? [closed]

I read the the Quantile Function commonly known as q-function is the inverse of the CDF for a normal distribution. $Q(x) = 1 - Q(-x) = 1 - \Phi(x)\,\!,$ where Φ(x)...
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Quantile function of Singh-Maddala

I am trying to understand how to derive the quantile function from the cdf. The Singh-Maddala cdf is Should i just solve for x ? Thank you
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Help me understand the quantile (inverse CDF) function

I am reading about the quantile function, but it is not clear to me. Could you provide a more intuitive explanation than the one provided below? Since the cdf $F$ is a monotonically increasing ...
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322 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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Calculating distribution minimum and maximum values from known p5/mode/p95 values

I am defining triangular and Beta-Pert distributions in MATLAB to produce random samples for Monte Carlo analysis. This is a trivial task if the minimum, maximum and mode are known using: makedist('...
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How to generate a more accurate distribution of sample of random variates?

I want to generate a sample of random variates choosing 5 out of 7 days of the week, $X_1,\ldots,X_5$, such that the aggregate counts resemble a given day-of-week profile, namely $$\mathbb{E}[\mathbb{...
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1answer
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Generation of Normal Distributed Numbers (with Box-Muller method?)

I want to generate several random, normal distributed numbers. At the moment I use the Box-Muller method. I have a function that returns a single number, and for this I use the following formula: ...
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1answer
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How to generate conditioned random variables from a density function?

I want to generate random variables from a distribution function using inverse sampling with the additional condition that the sampling should be conditioned, i.e., random generated variables should ...
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How does the inverse transform method work?

How does the inversion method work? Say I have a random sample $X_1,X_2,...,X_n$ with density $f(x;\theta)={1\over \theta} x^{(1-\theta)\over \theta}$ over $0<x<1$ and therefore with cdf $F_X(x)=...
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1answer
656 views

Linear Properties of the Quantile Function

Suppose $X$ is a random variable with continuous distribution function $F(x)$ and quantile function $Q_X(p)$ and let $Y = aX + b$ for some constants $a > 0$ and $b$. How can I prove that $Q_Y(p) = ...
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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 $$H(z)=L^{-1}_z(0.5)=\underset{...
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1answer
876 views

Conditional Expectation via Integral over Quantile Function

Following this thread "Does a univariate random variable's mean always equal the integral of its quantile function?" I tried to do a similar thing for a conditional expectation. It seems like my ...
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Does a univariate random variable's mean always equal the integral of its quantile function?

I just noticed that integrating a univariate random variable's quantile function (inverse cdf) from p=0 to p=1 produces the variable's mean. I haven't heard of this relationship before now, so I'm ...
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3answers
744 views

CDF raised to a power?

If $F_Z$ is a CDF, it looks like $F_Z(z)^\alpha$ ($\alpha \gt 0$) is a CDF as well. Q: Is this a standard result? Q: Is there a good way to find a function $g$ with $X \equiv g(Z)$ s.t. $F_X(x) = ...