Linked Questions
19 questions linked to/from Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X
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$Y=F(X) \implies Y\sim U(0,1)$? [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 ${\displaystyle F_{X}}$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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))...
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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 \...
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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 ...
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Calculate $\mathbb{E}[N(X)]$, where $N(·)$ is the cdf of the standard normal distribution, and $X$ is a standard normal random variable.
I'm stuck with this problem: Calculate $\mathbb{E}[N(X)]$, where N(·) is the cdf of the standard normal distribution, and X is a standard normal random variable.
Here's where I'm stuck: We know that:
$...
user1146971
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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 ...
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Show that $ \int_\mathbb{R} 2F(x)dF(x)=1 $.
Let $F$ be Stieltjes measure function (nondecreasing and right continuous) and $\mu$ be corresponding measure on $(\mathbb{R}, \mathcal{R})$. Assume that $F: \mathbb{R}\to [0,1]$ is continuous. Show ...
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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 $Y$...
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