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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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Convexity of a function of generalized inverse CDF.

How can I prove (or disprove) that the following function is convex on $X$. $$\rho(X,Z) = Max \{ F^{-1}_Z(t)-F^{-1}_X(t),0 \},$$ where $F^{-1}_X(t)= inf \{ x : F_X(x) \geq t \}$ with $0 \le t \le 1$....
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132 views

If $(Q_3-Q_2)=\frac34(Q_2-Q_1)$, then

If $(Q_3-Q_2)=\frac34(Q_2-Q_1)$, then There are more data which are less than the median value There are more data which are less than the modal value There are less data greater than the mean value ...
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29 views

Primitive of a function

I have a function $h_{j}(u_j) = 1-z^2 _j $ with $z_j= \Phi^{-1}(u_j)$, $\Phi^{-1}$ is the standard normal quantile function and $u_j \in (0,1)$ I want to show that $ \int_0 ^{u_j} h_{j} (\lambda) \, ...
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20 views

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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17 views

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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1answer
33 views

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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21 views

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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29 views

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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1answer
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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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37 views

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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30 views

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
21 views

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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66 views

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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26 views

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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1answer
53 views

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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1answer
49 views

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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30 views

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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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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117 views

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
62 views

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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1answer
48 views

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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78 views

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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115 views

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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56 views

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
58 views

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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171 views

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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1answer
308 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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107 views

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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38 views

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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551 views

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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1answer
119 views

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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1answer
133 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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82 views

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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2answers
79 views

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{...