# Questions tagged [quantile]

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

84 questions
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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### 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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### Bootstrap estimation of the 95% confidence intervals for the 95% quantile for gamma distribution

I cant find any where information or algorithm how to apply in steps the bootstrap procedure to estimate the 95% confidence intervals for the 95% quantile from a random sample. Does anyone knows how ...
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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 $\int_0 ^{u_j} h_{j} (\lambda) \, ... 1answer 20 views ### Tail difference of quantiles of (symmetric) distribution functions Assume, for example,$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 ... 1answer 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,... 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 ... 0answers 14 views ### 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 ... 1answer 976 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 ... 0answers 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 ... 1answer 1k views ### Sample median of Cauchy distribution is consistent. How? When we use chebyshev's inequality to show whether an estimator is consistent or not, we require the mean square error of the estimator and I do not know sample median's probability distribution. So ... 0answers 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 ... 1answer 33 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 ... 0answers 14 views ### Understanding the Von Mises quantile function + 95% CI? Given the following ts: ... 0answers 58 views ### 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\}.$$Define the ... 0answers 18 views ### How to calculate quantiles of sums? Let X_1, X_2 be independent normal distributions. Consider the quantile x such that P(X_1 + X_2 \le x) = \alpha for some \alpha \in (0,1). My question is, how does this quantity relate to the ... 0answers 37 views ### Calculate Quantile function Q(p) for the F(x) Given F(x) = 1/26(2x^2 + x - 10) and I need to find the Quantile function of Q(60). I have tried with Q = F^-1(p) but still, I have not got the correct answer. Note: Correct answer is 3.3364 0answers 31 views ### Inverse distribution function (quantile function) From Wiki: If the CDF ''F'' is strictly increasing and continuous then F^{-1}( p ), p \in [0,1], is the unique real number x such that F(x) = p . In such a case, this defines the "inverse ... 1answer 54 views ### How to show that \Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}}) In the middle of some proof, I have faced an expression \Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}}), where \Phi(\cdot)^{-1} is a quantile function of the standard normal distribution and x \in (0,1). ... 1answer 48 views ### Quantiles - supremum and infimum How to prove that$$\inf\{x \in \mathbb{R}: \mathbb{P}(X \le x) > \alpha \}=\sup \{x \in \mathbb{R}: \mathbb{P}(X<x) \le \alpha \}$$for any random variable X and \alpha \in (0,1)? 1answer 168 views ### How do you solve for lambda in an exponential distribution? I am working on a problem and am unsure how to solve it. The problem: Find an exponential distribution such that P(Z \ge 3) = .04 What I have done so far: P(Z\ge3) = 1 - P(Z\lt 3) We are ... 1answer 21 views ### How do you solve for the mean in a Normal Distribution? I am working on a problem and am a little bit stuck on how to solve it. The problem: Find a Normal Distribution with SD 2.5 and 5% Quantile at -15.2. What I have done so far:$$X=\mu+2.5Z.05=P(\... 0answers 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? $$\min_{\mu}\sum_{i=1}^n|y_i-\mu|=\text{median}\{y_1,\cdots,y_n\}$$ 2answers 33 views ### Choosing an interval of the CDF to find each quartile I have a random variable$X$which has the following CDF: $$F(y) = \left\{\begin{array}{ll} 0 & : y \lt 0\\ \frac{y}{30} & : 0 \le y \lt 20\\ \frac{2}{3} + \frac{y-20}{60} & : 20 \le y \lt ... 0answers 29 views ### find scale and shape parameter of Weibull distribution by having the 50 and 90 percent quantiles I have the 50 and 90 percent quantile of a Weibull distribution. Is it possible to get the shape and scale parameter just with this information? Mathematically I have this formula for calculating the ... 0answers 40 views ### Pushforward-change-of-variable with quantile function I’ve been dealing with an issue about change-of-variable formula. Let \mu be a probability measure on \mathbf R_+. Let F(x) = μ([0,p]) and Q its quantile function, ie Q(p) = \inf \{q \in \... 1answer 26 views ### Implications on the cdf of \epsilon_1-\epsilon_2 of conditions on the cdf's of \epsilon_1, \epsilon_2 Consider three random variables \epsilon_1, \epsilon_2, X. Let F_{\epsilon_i}(\cdot| x) denote the cumulative distribution function (cdf) of \epsilon_i conditional on X=x for any i\in \{1,2\}... 1answer 53 views ### Uniform transformation of a quantile Let x_{\alpha} = \inf \{x \in\mathbb{R}: F_X(x) \geq \alpha\}, U \sim Uniform(0,1) and Z=x_{U}. I need to prove that Z has the same distribution as X. Obviously this is true as can easily be ... 1answer 49 views ### Inequality involving quantiles Suppose that X and Y are r.v.s such that F_X (the cdf of X) is continuous and$$ \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+\... 0answers 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 ... 2answers 553 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] ... 1answer 118 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 ... 1answer 196 views ### Prove X(\omega) = \sup\{y \in \mathbb{R}: F(y) < \omega\} is a random variable. Let F be a distribution function. On (\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda) where \lambda denotes Lebesgue measure. Define X: \Omega \to \mathbb{R} by X(\omega) = \sup(y \... 1answer 66 views ### What are the advanatages of CDFs for RNG over simple random sampling? If I understand Cumulative Distribution Functions (CDFs) correctly, they can be used for random number generation from a given dataset as follows: Build a CDF that maps data points to an ordinal ... 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 ... 5answers 6k views ### Are there 3 or 4 quartiles? 99 or 100 percentiles? So I understand that a quartile is a quantile where the data is divided into four groups. 1 2 3 ---|---|---|--- And 1, 2, and 3 are the quartiles. The ... 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?... 1answer 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: ... 1answer 116 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 ... 2answers 33 views ### 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/... 1answer 57 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 ... 0answers 27 views ### Evaluation of pth 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 pth Quantile, for which we have to set \int_{... 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 ... 0answers 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 ... 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 ... 0answers 42 views ### 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 ... 2answers 185 views ### 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 ... 0answers 80 views ### 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$...
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 $f(x, \theta_n)$. ...