# 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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### Examples of distributions with easily solvable quantile functions but hard to solve CDFs

I'm interested in examples of probability distributions where the quantile function $F^{-1}(p)$ exists in closed form or is easy to calculate but where the cumulative distribution function (CDF) $F(x)$...
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### Does CDF must have value 0 at lowest possible input?

Suppose $F$ is the CDF of a real valued random variable. I know that $F(- \infty) = 0$, because the RV cannot take a value less than that. But I was thinking of an RV whose value for sure comes from, ...
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### Is it possible to use acritical value from the F distribution with a CI of 95% and obtain the z value with a CI of 99% in the same problem?

I am studying this article "Variance sensitive adaptive threshold-based PCA method for fault detection with experimental application (Alkaya,2011)" in order to apply the adaptive threshold ...
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### Variance of CDF F(x) when x follows the same probability law as F

The cumulative distribution function $F(x)=P(X\leq x)$ is a fixed probability number. I wonder what its variance is, if we let the argument be a random variable following the same distribution as $X$. ...
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### Cumulative distribution of Gaussian conditional independent random variables

Suppose X, Y, Z are three jointly Gaussian random variables and X and Z are independent given Y. For example, take three r.v. from a OU process. Here is some R code:...
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### Properties of the inverse normal cdf and permutation probabilities as models for horse racing

Let $T_i$ be the running time of horse $i$ and $T_i \sim N(\theta_i,1)$ and the $T_i$'s are independent. Then Henery (1981) showed that the probability $P(T_1<T_2<\cdots <T_n)$ can be ...
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### Analyzing the minimum of a sequence of iid random variables given the CDF [duplicate]

I'm doing a question about a sequence of i.i.d. random variables that all have CDF $F(x) = 2^\alpha(2 - x)^{-\alpha}$ for $x < 0$ and $F(x) = 1$ otherwise. Without asking the whole question, ...
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### Conditional CDF given one dimension equals derivative of joint CDF towards that dimension divided by the density at that dimension?

So I am familiar with the following: $$P\left(X<x|Y=y\right) =\int_{-\infty}^{x}f\left(X=u|Y=y\right)du=\frac{1}{f\left(Y=y\right)}\cdot\int_{-\infty}^{x}f\left(X=u,Y=y\right)du$$ But during a ...
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we have that $P(X \leq x, Y \leq y) = \int \int_{s \leq x, t \leq y} f_{X,Y}(s,t) dsdt$ But how would for example $P(X \leq x, Y \geq y)$ Be defined? Would it just be: $P(X \leq x, Y \geq y) = \int \... • 83 2 votes 1 answer 85 views ### CDF of$\max\$ under conditions

Let, $$g(\alpha,\beta) = \begin{cases} \frac{\alpha}{\beta}, & \text{if } \alpha > \beta \\ 0, & \text{if } \alpha \leq \beta \end{cases}$$ I want to find ...
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### Auxiliary randomization and the generalized CDF inverse

I'm trying to solve Homework 4 from professor Ryan Tibshirani's class on "Advanced Topics in Statistical Learning" [pdf] at UC Berkeley. It deals with basic facts about CDFs and quantiles. ...
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### Estimation of Distribution using multiple ECDFs

Every day, I keep track of the processing times for each input to my CPU and create empirical cumulative distribution functions (ECDFs) based on this data. Let's assume I have 100 observations per day ...
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