# Questions tagged [cumulative-distribution-function]

Cumulative distribution function. While the PDF gives the probability density of each value of a random variable, the CDF (often denoted $F(x)$) gives the probability that the random variable will be less than or equal to a specified value.

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### Copula between a distribution and its univariate transformation

I'm trying to compute the copula (or joint distribution) between x and a univariate transformation, like say sin(x). That is compute $C_{XY}$ (or $F_{XY}$) given that $x \sim U(0,1)$ and $y = sin(x)$ ...
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### Estimation of CDF in multiple points

Suppose we have a sample $X_1, \ldots, X_n$ of i.i.d. real-valued random variables with an (unknown cumulative) distribution $F$. The goal is to estimate the value of $F$ in multiple points. That is, ...
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### Variance of standard normal transformation of normal variable [duplicate]

Is there a closed-form solution for the variance of $Y = \Phi\left(X\right)$, where $X \sim N \left(\mu, \sigma^2\right)$ and $\Phi$ is the standard normal CDF? I can find the variance for some ...
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### Sum of indepedent random variables and a constant

Let $X_1$ and $X_2$ be independent Normal random variables with mean $\mu_1$ and $\mu_2$, and variances $\sigma_1$ and $\sigma_2$. Let $Y = X_2-X_1 + c$, where $c$ is a constant. For notational ...
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### Fairness metric computation

I am trying to implement a fairness metric (it is called Statistical Parity, Demographic Parity, Group Fairness,... depending on the website/paper): $$P(\hat Y|A=a)=P(\hat Y|A=b)$$ The idea is to ...
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### What is the distribution of the CDF of a sample drawn from a multivariate normal?

Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
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### Derivation and meaning of 1 minus the cumulative distribution?

If the cumulative distribution function of a random variable is $$F(x) = P(X\leq x)$$ how can this be transformed mathematically to, and the meaning of $$1-F(x)$$
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### Why is cumulative distribution function monotone non-decreasing?

If you have a quantity ${X}$ that takes some value at random, the cumulative distribution function ${F(x)}$ gives the probability that ${X}$ is less than or equal to ${x}$, that is: \begin{equation*} ...
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### Is it accurate to take the maximum distance between CDF and ECDF only at the edges? (Kolmogorov-Smirnov Test)

I have two samples, one obtained empirically and the other is the result of a simulation. I want to tune the simulation so that the result resembles the reality, for that I will minimize the KS ...