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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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### Define a region $R \subset \mathbb{R}^2$ such that $P(x \in R) = 0.95$

Consider a random vector $X \sim N_2(\mu,\Sigma)$. Define a region $R \subset \mathbb{R}^2$ such that $P(x \in R) = 0.95$ Hint: The $0.95-$quantile of $\chi^2$ is about $5.99$ . I think we are ...
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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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### 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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### 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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### Understanding the Von Mises quantile function + 95% CI?

Given the following ts: ...
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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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### 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 $x \in (0,1)$. ...
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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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### 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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### 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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### 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 ...