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Questions tagged [quantile]

one of several equally-frequent subranges of a data set or random distribution; for example, a percentile or quartile

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Tail difference of quantiles of (symmetric) distribution functions

Assume, for example, $z_\alpha$ are $\Phi^{-1}(\alpha)$ quantiles from standard normal distribution, $\alpha > 0$. If we are interested in the sum$$z_\alpha + z_{1 - \alpha}$$ for standard normal ...
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What is the lower quartile of the set of data?

I came across this problem asking the lower quartile of the ungrouped data. My answer is 3, but other references say it should be 2.5. Here's the data: 1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 8, 10, 11, 14, 15,...
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Why is quantile function of uniformly distributed random variable a random variable?

I have the quantilfe function $F^{-1}$ of a random variable which is defined as: $F^{-1}: ]0, 1[ \ni u \rightarrow F^{-1}(u) = inf\{x: F(x) \geq u\} \in \mathbb{R}$. Now I can define a new random ...
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Quantile function question

is there a way how to get quantile function from random variable $Y$ defined as: $$ Y = \begin{cases} 0 \text{ ... with probability 0.25}\\ f(x) \text{ ... with ...
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Taylor series expansion of quantile function

Suppose $Y$ and $Z$ two random variables, $\lambda \in \mathbb{R} $. We note $F^{-1}_{Y}(\alpha)$ the quantile function of the variable $Y$ at the quantile level $\alpha \in (0,1)$. Do you have any ...
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How to calculate the area difference of an integral cut at a line with slope?

Using trapezoidal method (because I have a vector and no function is available) I know how to calculate the area difference one gets when the integral (Fig. 1) is cut by a horizontal line such as ...
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28 views

Quantile-Quantile (QQ) Plots

I understand how to assess whether an exponential/normal distribution is suitable to model a piece of data when the parameters are given, i.e. finding the theoretical quantiles and plotting against ...
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For cdf $F(x)$ ad empirical cdf $F_n(x)$, show that $|F^{-1}\big(F_n(\xi_p)\big)-\hat\xi_p|\overset{a.s}\to0$

Suppose $X_1,\cdots,X_n$ are i.i.d. continuous random variables from distribution with cdf $F(x)$.Let $F_n(x)$ be a random variable defined by $$F_n(x)=\frac{1}n\sum_{i=1}^nI\{X_i\le x\}.$$ Define the ...
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How to calculate quantiles of sums?

Let $X_1, X_2$ be independent normal distributions. Consider the quantile $x$ such that $P(X_1 + X_2 \le x) = \alpha$ for some $\alpha \in (0,1)$. My question is, how does this quantity relate to the ...
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Calculate Quantile function Q(p) for the F(x)

Given F(x) = 1/26(2x^2 + x - 10) and I need to find the Quantile function of Q(60). I have tried with Q = F^-1(p) but still, I have not got the correct answer. Note: Correct answer is 3.3364
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Inverse distribution function (quantile function)

From Wiki: If the CDF ''F'' is strictly increasing and continuous then $ F^{-1}( p ), p \in [0,1], $ is the unique real number $ x $ such that $ F(x) = p $. In such a case, this defines the "inverse ...
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Quantiles - supremum and infimum

How to prove that $$\inf\{x \in \mathbb{R}: \mathbb{P}(X \le x) > \alpha \}=\sup \{x \in \mathbb{R}: \mathbb{P}(X<x) \le \alpha \}$$ for any random variable $X$ and $\alpha \in (0,1)$?
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How do you solve for lambda in an exponential distribution?

I am working on a problem and am unsure how to solve it. The problem: Find an exponential distribution such that P(Z $\ge$ 3) = .04 What I have done so far: P(Z$\ge$3) = 1 - P(Z$\lt$ 3) We are ...
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1answer
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How do you solve for the mean in a Normal Distribution?

I am working on a problem and am a little bit stuck on how to solve it. The problem: Find a Normal Distribution with SD 2.5 and 5% Quantile at -15.2. What I have done so far: $$X=\mu+2.5Z$$ $$.05=P(\...
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Prove that $\min_{\mu}\sum_{i=1}^n|y_i-\mu|=\text{median}\{y_1,\cdots,y_n\}$ [duplicate]

How to prove the equation below in a simple way? $$\min_{\mu}\sum_{i=1}^n|y_i-\mu|=\text{median}\{y_1,\cdots,y_n\}$$
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Choosing an interval of the CDF to find each quartile

I have a random variable $X$ which has the following CDF: $$F(y) = \left\{\begin{array}{ll} 0 & : y \lt 0\\ \frac{y}{30} & : 0 \le y \lt 20\\ \frac{2}{3} + \frac{y-20}{60} & : 20 \le y \lt ...
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find scale and shape parameter of Weibull distribution by having the 50 and 90 percent quantiles

I have the 50 and 90 percent quantile of a Weibull distribution. Is it possible to get the shape and scale parameter just with this information? Mathematically I have this formula for calculating the ...
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Pushforward-change-of-variable with quantile function

I’ve been dealing with an issue about change-of-variable formula. Let $\mu$ be a probability measure on $\mathbf R_+$. Let $F(x) = μ([0,p])$ and $Q$ its quantile function, ie $Q(p) = \inf \{q \in \...
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Implications on the cdf of $\epsilon_1-\epsilon_2$ of conditions on the cdf's of $\epsilon_1, \epsilon_2$

Consider three random variables $\epsilon_1, \epsilon_2, X$. Let $F_{\epsilon_i}(\cdot| x)$ denote the cumulative distribution function (cdf) of $\epsilon_i$ conditional on $X=x$ for any $i\in \{1,2\}$...
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Uniform transformation of a quantile

Let $x_{\alpha} = \inf \{x \in\mathbb{R}: F_X(x) \geq \alpha\}$, $U \sim Uniform(0,1)$ and $Z=x_{U}$. I need to prove that Z has the same distribution as X. Obviously this is true as can easily be ...
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Inequality involving quantiles

Suppose that $X$ and $Y$ are r.v.s such that $F_X$ (the cdf of $X$) is continuous and $$ \sup_{r\in\mathbb{R}}|F_X(r)-F_Y(r)|\le \epsilon. $$ Is it true that $\mathsf{P}(X\le q_Y(\alpha))\le \alpha+\...
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Asymptotic behaviour of the process $U_n=U_{n-1}+s(U_{n-1})X_n$, where $X_n$ is iid

Let $s, s^*:\mathbb{R}^+ \to \mathbb{R}^+$ ($0\in\mathbb{R}^+$) such that $0 \le s(x), s^*(x) \le x$ for every $x \in \mathbb{R}^+$ and $X, X_1, \ldots$ is iid with $\mathbb{E}X>0$. Let $U_{0,s}=1$ ...
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How do i explain (using visual plots) the deviation of my empirical data from normality?

I have made a qqplot as well as a pdf of my random variables (see attached figure). Looking at the ...
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1answer
854 views

Quantile function for binomial distribution?

A test will succeed with a certain percentage. Now this test is repeated X number of times. I want to be able to get an estimate of the total number of succeeded test. Given that I know both the ...
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Where can I find examples of Skorokhod representations?

So, I recently (re-)discovered that random variables learned in elementary probability such as the exponentially distributed random variable $X$ with cdf $F_X(x) = 1-e^{- \lambda x}$ can be explicitly ...
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1answer
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Explicitly representing a random variable such as $ X(\omega):=\frac{1}{\lambda} \ln \frac{1}{1-\omega}$, which is exponential

Previously: (Dumb question: Computing expectation without change of variable formula) I was wondering how to compute $E[X]$ by $\int_{\Omega} X d\mathbb P$ rather than $\int_{\mathbb R} x d \mathcal ...
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Expected value of position in subset.

I have set of $n$ elements $[0, ... n-1]$. I randomly pick a subset $S$ of $k$ elements (also ordered). Assume I have $t \in \{1, .. k\}$. What is expected value of $t$-th position in ordered subset?...
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How is this statistical index called?

I need to calculate an index that is to be derived like this: If we have some data: 850 700 500 480 300 100 50, we first sort it from the large to small: ...
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Deduce the variance of a normal distribution from its $2\%$ and $95\%$ quantiles

This is a Cambridge A Level Question that I am currently trying to solve: Metal rods produced by a machine have lengths that are normally distributed. It is known that 2% of the rods are rejected ...
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If parametric quantile esimation estimates $p$ by computing the MLE, then how to get non-parametric $p$? [closed]

For non-parametric or parametric quantile estimation. If parametric quantile esimation estimates $p$ by computing the MLE, then how to get non-parametric $p$? Related: https://mathoverflow.net/...
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A question about quantile function of normal distribution

Suppose that I know a normal distribution of randome varible $X$ satisfies the property that $P(20<X<30) = 0.9$, is that true that the lower quartile of $X$ must be between $20$ and $25$? I ...
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Evaluation of $p$th Quantile?

Let $X$ be a random variable having PDF $f(x) = \frac{1}{\lambda} e^{-\frac{\lambda}{x}} , x > 0 , \lambda > 0$. And I was trying to find out the $p$th Quantile, for which we have to set $\int_{...
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1answer
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Calculate t-quantiles and $\chi^2$-quantiles

How are t-quantiles and $\chi^2$-quantiles actually calculated? I find it difficult to find a formula. For example, the t-quantile for 0.975 and 50 degrees of freedom is approximately 2. This is ...
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Quantiles of a Brownian Motion

I am looking for the p-th percentile of a stochastic process $X_t$ that satisfies $dX_t = \mu(X_t) dt + \sigma(X_t) dW_t$ where $W_t$ is a standard Brownian motion. I believe that the p-th percentile ...
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303 views

What is meant by the Fisher information of a particular of a particular quantity for a quartile function?

My provided definition of the Fisher information $\mathcal{I}(\theta)$ is the expected value of the observed information $I(\theta)$, where $I(\theta)$ is the second derivative of the log-likelihood ...
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Estimating Quantile Function Using Pseudo Sample

Let $\{X_i\}_{i=1}^n$ be a random sample, $Y_i=\delta(X_i)$ for some function $\delta(\cdot)$, and $Q(\tau)$ be the population quantile function of $Y_i$. We can estimate $Q(\cdot)$ by the empirical ...
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Bootstrap method & Confidence Interval

I'm trying to figure out how this method works. My data: 1000 samples from unknown distribution. I need to create 40 vectors from those 1000 samples (each vector with 20 samples) For every one of the ...
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How to efficiently find quantile (root) of integral function

Let $f(x)$ be a density function on $\mathbb R$; I want to find numerically the $\alpha$ quantile of the associated distribution, i.e. I want to find $c$ such that $$\int_{-\infty}^c f(x)dx = \alpha$...
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Basic Quantile Calculation

I have a slight confusion with the current method of calculation of a quantile for a give ungrouped distribution. To give you an example, i shall refer to calculation of a Quartile, but this doubt ...
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Quantile function of the r th order statistics of i.n.n.i.d. random variables

I am repeatedly drawing $n$ random samples where the first sample is drawn from $f(x, \theta_1)$ the second from $f(x, \theta_2)$, $\enspace \dots$ , and the $n^{\text{th}}$ from $f(x, \theta_n)$. ...
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consistency of inverse cdf

If $\hat{F}(x)$ is a consistent estimator of $F(x)$ where $F(x)$ is a cdf, can we state that $\hat{F}^{-1}(x)$ is also a consistent estimator of $F^{-1}(x)$? Is that straightforward? Why? You can make ...
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Construct a random variable with a given distribution

Suppose that ($\Omega$, $\mathcal{F}$,$P$) where $\mathcal{F}$ is the $\sigma$-algebra of Lebesgue measurable subsets of $\Omega\equiv[0,1]$ and $P$ is the Lebesgue measure. Let $G:\mathbb{R}\to[0,1]$ ...
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Sample quantile of order p

My book says the following : "Let $X_{(1)}, X_{(2)}, ..., X_{(n)}$ be a set of values ordered in ascending order ($X_{(1)} \leq X_{(2)} \leq ... \leq X_{(n)})$. For a given $p$ ($0 \le p \le 1$), ...
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How to prove that sample of size $O(\epsilon^{-2} log \delta^{-1})$ is enough to predict quantiles?

There is a known problem: You are given a stream of numbers and you need to find it's $q$-th quantile ($0 \le q \le 1$). You may get wrong answer but you need to return answer between $q-\epsilon$-th ...
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127 views

CDF and Quantile Function

Assume that $F_X$ is the CDF of the random variable X and $Q_X$ its quantile function.Prove that $Q_X(F_X(t)) \le t $ and $F_X(Q_X(p)) \ge p $. I substituted $F_X(t)=P[X \le t]$ and then substituted ...
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Upper bound for distance between actual and sample quantiles

Let $\xi_p$ be pth quantile of the distribution $F(x)$ with derivative at $\xi_p$, $f(\xi_p) >0$. Then, $$ |\hat\xi_{p,n} - \xi_p| \leq \frac{2}{f(\xi_p)}\sqrt{\frac{\log n}{n}} $$ almost surely ...
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basic Quantile proves

Let this be my definition of a quantile funktion. X is a real-valued random variable. And let F be it's distribution function. then \begin{align*} F^{-1}(a):=\inf\{x\in \mathbb{R}: F(x) \ge a\}. \end{...
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1answer
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Calculating Quantile for a specific problem

so I understand how to calculate the 0.5 quantile of the given question. I calculate the CDF of x and then I multiply it to 0.5 But what if there's more than one function for multiple intervals? How ...
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1answer
127 views

Inverse CDF of a function

I am trying to find the Inverse CDF (quantile function) of this function to create an random number generator: $f(p_a) = (\beta + 1) p_a^{\beta} \text{ where } \beta \geq 1 \text{ and } 0 \leq p_a ...