Linked Questions

Finding density function, plus showing $X \sim F$ where F is cdf of X, $X = F^{-1}(U)$, $U\sim unif(0,1)$ [duplicate]

Suppose $X$ has a continuous, strictly increasing cdf $F$. Let $Y = F(X)$. What is the density of $Y$? Then let $U \sim unif(0,1)$ and let $X = F^{-1}(U)$. Show that $X\sim F$. The first part seems ...

Craig's user avatar

  • 543

$Y=F(X) \implies Y\sim U(0,1)$? [duplicate]

I have a doubt on this theorem: The probability integral transform states that if ${\displaystyle X}$ is a continuous random variable with cumulative distribution function ${\displaystyle F_{X}}$, ...

Relure's user avatar

  • 4,051

2 votes

1 answer

1k views

What is the intuitive explanation for the CDF of any random variable to follow uniform distribution (0,1)? [duplicate]

If $X$ is a continuously distributed random variable and $F$ is the c.d.f. of its distribution, then $F(X)$ is uniformly distributed in the interval $(0,1).$ While i'm clear with the mathematical ...

LUCIFER's user avatar

  • 109

Need help in computing the p.d.f. of the random variable $F[X]$ [duplicate]

Let $X$ be a continuous random variable with probability density function, $f$ and c.d.f $\;F$. Suppose that $f$ is continuous and $f(x)>0$ for all $x\in \mathbb{R}$. Compute the p.d.f. of the ...

Shweta Aggrawal's user avatar

  • 5,373

0 votes

0 answers

37 views

Uniform Probability distribution question [duplicate]

Suppose $X$ is a continuous random variable with a cumulative distributive function of $F(x)$. Let $Y$ be a variable such that $Y$ can be represented in terms of $X$, i.e $Y=F(X)$. Show that the ...

Quippy's user avatar

  • 1,261

3 votes

3 answers

2k views

Given the cumulative distribution function find a random variable that has this distribution.

We are given a Cumulative distribution function $(CDF)$ how do we find a random variable that has the given distribution? Since there could be a lot of such variables we are to find anyone given that ...

aroma's user avatar

  • 440

Proof of $Y=F_X(X)$ being uniformly distributed on $[0,1]$ for arbitrary continuous $F_X$

This question is related to Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X, with the difference being that $F_X$ (the probability distribution function of ...

syeh_106's user avatar

  • 2,814

4 votes

0 answers

2k views

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$

Let $X$ be a continuous random variable with cdf $F$. Show that $Y = F(X)$ has uniform $(0,1)$ distribution and therefore $X = F^{−1}(Y)$. My Sol: $P(Y \leq y ) = P(F(X) \leq y) = P\left(F^{-1}(F(X))...

Alex Chavez's user avatar

  • 884

Show that this random variable is uniformly distributed in $\left(0,1\right)$

Let $X$ be a continuous random variable with distribution function $F_{X}\left(x\right)$ and let $Y=F_{X}\left(x\right)$, show that $Y\sim U\left(0,1\right)$, where $U$ is the uniform distribution. ...

eNR's user avatar

  • 321

0 votes

2 answers

509 views

Function of random variable has uniform distribution [closed]

For any random variable $X$ taking values in $[0,1]$ with distribution $\mu_X$, and $f$ defined by $f(t) = \int_{-\infty}^t d\mu_X$, I'm stuck trying to show that $$P(f(X) \leq a) = a$$ for all $a \...

user435571's user avatar

  • 695

Product of two uniform random variables/ expectation of the products

Suppose I want the expectation, $E\Phi(X-\mu)\Phi(\mu-X)$, where $\Phi(.)$ represents the Normal CDF, and X is $Normal(\beta,1)$. Consequently $\Phi(.)$'s are uniform[0,1] and at the same time two ...

Shakil's user avatar

  • 101

1 vote

1 answer

214 views

Rigorous proof of CDF $F_X(X)$ is uniform

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be given. It is well known that, for continuous random variable $X(\omega)$, with CDF $F(x)$, then $F(X(\omega))$ is a uniform random variable. I want to look ...

Violapterin's user avatar

  • 1,645

0 votes

2 answers

169 views

Distribution of Y=F(X) where F is the cdf of continuous X and X is a Multivariate r.v.

Here is a well known fact (from this question): Let $X$ be a random variable (r.v.) with a continuous and strictly increasing c.d.f. function $F$. Define a new random variable $Y$ by $Y=F(X)$. Then $Y$...

StevenG's user avatar

  • 123

Distribution function of a random variable defined by a cumulative distribution function [duplicate]

I have been given a question: Suppose that $F(x)=P(X\leq x)$. Show that if $F$ is continuous and $Y:=F(X)$, then $P(Y\leq y)=y$ for all $y\in [0,1]$. I have been having difficult even starting to ...

Keen-ameteur's user avatar

  • 5,611

Tranform normal distribution to be over [0,1]

For Quasi Monte Carlo, given the standard normal distribution $$ f(x) = \frac{1}{\sqrt{2\pi}}\exp{\frac{-x^{2}}{2}} $$ over the real line ($\mathbb{R}$), how can this be transformed to be a density ...

Abby's user avatar

  • 177