# Questions tagged [quantile-function]

For questions related to the so called quantile function of a cumulative distribution function or generalized inverse of a cumulative distribution function.

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### Adequate Root Finder To Compute The Quantile Function

The cumulative distribution function of the standard normal distribution $\Phi(z)=\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^z e^{-t^2/2}dt$ cannot be expressed in terms of elementary functions, ...
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### Prove the equality: $\sup\{x\geq0 : F_X(x)\leq t\}=\int_0^{+\infty} 1_{\{ F_X(x)\leq t\}}\,dx$. [closed]

I was trying (unsuccessfully) to prove the following equality $$\sup\{x\geq0 : F_X(x)\leq t\}=\int_0^{+\infty} 1_{\{ F_X(x)\leq t\}}\,dx.$$ Can anyone give me an hint?
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### Function of quantile function

I stumbled across this Theorem in a paper, but I am not able to obtain its reference or solve it. Please help. Let $X_1, ..., X_n$ be independent random variables with CDF $\Phi_1,...,\Phi_n$ (which ...
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### List of closed form special cases and transformations of Wolfram language’s inverse beta regularized $\text I^{-1}_x(a,b)$.

The Wolfram Language’s Inverse Beta Regularized $\text I^{-1}_z(a,b)$ is a quantile function. This applicable yet obscure function appears in Excel as BETA.INV and a special case of it as the Inverse ...
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### $\mathcal{N}(0,1)$ c.d.f. and quantile

It is often useful to estimate the quantile for $\Phi^{-1}(d)$ for some real number $d$. Here $\Phi\sim \mathcal{N}(0,1)$. We cannot use for some unknown reasons the Taylor expansion as for estimating ...
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### Quantile function, bivariate joint density.

Consider random variables $X,Y$ with joint density function $$f(x, y)=\frac{1}{2 \pi}\left(1+x^2+y^2\right)^{-3 / 2}$$ I want to find the quantile function of $|Y|$. I have learned how to find the ...
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### Question about the range of the quantile function

Let $X$ be a random variable with probability distribution $F$. To define the question. We have to define to notions. First, we define the points of support of $X$ as \begin{align} {\rm supp}(X)= \{x: ...
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### Question on definition for proving a theorem regarding simulation.

Theorem. Suppose the random variable $U$ has a uniform $(0,1)$ distribution. Let $F$ be a continuous distribution function. Then the random variable $X=F^{-1}(U)$ has distribution function $F$. In the ...
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### Convergence of $p^{\text{th}}$ quantile estimator for sample from Exponential distribution

Convergence in probability of $p^{\text{th}}$ quantile estimator for iid sample $X_1, \ldots X_n$ from Exponential distribution given by $f(x, \lambda) = \lambda e^{-\lambda x}$ The $p^{\text{th}}$ ...
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### If $\mu$ is atomless then $F \circ F^{-1} (t) = t$ for all $t \in (0, 1)$

I'm trying to prove this property. Could you have a check on my attempt? Let $\mu$ be a Borel probability measure on $\mathbb R$ and $F$ its c.d.f. Then $F$ is right-continuous and non-decreasing. ...
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### Quantile function: $F(x) \ge t \iff x \ge F^{-1}(t)$ is not true if $F^{-1} (t) := \inf \{x \in \mathbb R \mid F(x) \color{red}{>} t\}$?

I'm reading about quantile function of a probability measure. However, I could NOT prove the property (c) $F(x) \ge t \iff x \ge F^{-1}(t)$ if $F^{-1}$ is defined with the strict inequality, i.e., \...
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### Integral of Quantile Function is Expected Value?

Take a distribution function $F$. If $Q_F : [0, 1] \rightarrow \mathbb{R}$ is defined by $Q(p) = \inf\{x | p \leq F(x)\}$, is it true that $$E_F[X] = \int_0^1 Q_F d\mu$$ where $\mu$ is the Lebesgue ...
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Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of real-valued random variables. In a proof of a theorem, the author starts with $$\lvert Cov(X_i,X_j)\rvert\leq \int_0^{\alpha_{\lvert i-j\rvert}}[Q_i(u)]^2+... • 2,689 0 votes 1 answer 320 views ### Integrals of quantile function Suppose X = [0,1] and we have two cumulative distribution functions F and G on X of two random variables. I've just read the claim$$\int_0^1 |F^{-1}(x) - G^{-1}(x)|dx = \int_0^1|F(y) - G(y)|...
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For a random variable defined by the PDF $f(x)$ and CDF $F(x)$, Characteristic function $CF(x)$ (if it exists) is given by the Fourier transform of $f(x)$. Quantile $Q(x)$ (if it exists) is given by ...