Questions tagged [cdf]

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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Random variable without pdf but with a cdf?

In this video, the professor says that some random variables have no pdf but do have a cdf. Also, in my course material, I studied that converging in mean was stronger than converging in cdf which ...
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Can't compute CDF for Inverse Gaussian distribution

I am trying to implement in Python the CDF of the Inverse Gaussian distribution: Inverse Gaussian pdf: $$ f(x) = \sqrt{\frac{\lambda}{2\pi x^3}}e^{-\frac{\lambda(x-\mu)^2}{2\mu^2x}} $$ Inverse ...
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How to estimate mean and variance for rate of change when I only have state data at different ages

I'll give you the intuition behind my problem first. I have data on whether children ($n \approx 200$) can read and their age in integers from 0 to 14. For each age, it is straightforward to ...
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Putting into words variable transformation

If Y is a random variable which comes from a transformation of X : $ Y = \phi(X) $, the formula for the tranformation of this random variable can be written as follows : $$ F_Y(y) = P(Y \leq y) = P(X \...
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Intermediate proof for Glivenko-Cantelli Theorem

Show that for any cdf, the following holds: $sup_{x\in \mathbb{R}}|F_n(x)-F(x)|\leq sup_{u\in [0,1]}|\overline{F}_n(u)-F(u)|$ Where $F_n:=\frac{1}{n}\sum_{i=1}^n1_{(-\infty,x]}(X_i)$ is the empirical ...
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Can we sample from both pdf and cdf?

my question is quite generic. I am currently studying the algorithms calculating random numbers from distributions: In inverse transform method we get the cumulative distribution function in the end ...
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Help me understand how the following likelihood function is derived

A week ago, I asked a question concerning the Taylor expansion of an arbitrary distribution function. As noted by a member of the forum, the question was vague and perhaps incorrect. I had asked this ...
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How is an 'ogival function' defined?

Reading on a paper on factor analysis and measurement invariance I find the description of some functions as 'ogival' functions. In Google I find it referenced mostly in papers from the '70s and '80s....
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Prove that $P(X \le a) + P\{Y \le \frac{1}{a}\} = 1$

Prove that if $X$ has the F-distribution with $(m, n)$ d.f. and $Y$ has the F-distribution with $(n, m)$ d.f., then for every $a > 0$, $$ P(X \le a) + P\left\{Y \le \frac{1}{a}\right\} = 1 $$ I ...
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CDF*[1-CDF]/PDF — name? integrable?

Suppose I have a random variable $X\in\mathbb R$ distributed according to a smooth nonzero probability density function (PDF) $f(x)$, with cumulative distribution function (CDF) $F(x):=\int_{\infty}^x ...
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About unique determination of symmetric point (or center) of a distribution based on pdf or cdf

Suppose we have a distribution that is known to be continuous and symmetric, and is otherwise unknown. We want to decide whether it is actually centered at zero using an equation involving pdf or cdf. ...
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How to prove that a function is 2-increasing (copula)

There are three conditions to prove that a function is a copula: $C(u,0)=0=C(0,v)$ grounded. $C(u,1)= u, C(1,v)= v$. $C(u,v)$ 2-increasing function. Here I am concerning in the last condition how to ...
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Is the definition of symmetric distribution using cdf correct?

Based on wikipedia (https://en.wikipedia.org/wiki/Symmetric_probability_distribution), a distribution is symmetric about $x_0$ if and only if it is a distribution whose pdf(or pmf) $f(\cdot)$ ...
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point such that area to the right of that point under one gaussian is 5% of area under a second gaussian

Say I have two gaussian random variables $Z_1 \sim f_1 = f(\cdot|\mu_1, \sigma_1)$ and $Z_2 \sim f(\cdot|\mu_2, \sigma_2) = f_2$, where $f$ is the gaussian density. How can I calculate the value of $x$...
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How to compute the median of a continuous distribution?

I don't have a solid background in statistics so the concept of probability density functions in the statistics course I'm taking is new to me. I need to derive the median of a continuous distribution ...
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Creating a Probability Plot of a Custom Distribution

Let's say we have some icdf function, which I will paste below: ...
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Which $\mu$ hold so that integral of CDF (from $\mu$ to $\infty$) equals to integral of 1-CDF (from $-\infty$ to $\mu$)?

What is the $\mu$ s.t. $$\int_{\mu}^{\infty}1-F(x)dx = \int_{-\infty}^{\mu}F(x)dx?$$ Here $F(x) = P(X\leq x).$ Should $\mu$ be the median of X, i.e. $0.5=F(\mu)$? I think $\mu$ should be the point so ...
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Copula between a distribution and it's univariate transformation

been reading for a while, first time questioner. I'm trying the compute the copula (or joint distribution) between x and a univariate transformation, like say sin(x). That is compute $C_{XY}$ (or $F_{...
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What is the ppf of the truncated normal distribution?

What is the percent point function (ppf), or inverse cdf, of the truncated normal distribution? The distribution and cdf is defined here: https://en.wikipedia.org/wiki/Truncated_normal_distribution $$...
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Why is CDF of binomial random variable step function

I just read that the cumulative distribution function for a binomial random variable is a "step function where the function is flat and then jumps at each nonnegative integer value". Can ...
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Convergence in Distribution and Ordered Statistics [duplicate]

Let $X_1, X_2, \ldots$ be iid from Exp$(\theta)$ with density function $f(x) = \frac{1}{\theta}e^{-\frac{x}{\theta}}$. Find the limiting distribution of $M_n = Y_1 - \theta\ln(n)$ and $T_n = nY_n$, ...
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Inverse CDF (Quantile) of Piecewise Function [duplicate]

This question may be insanely simple, but I'm unsure. Let's say we have the following function: $$ f(x) = \begin{cases} x & 0 \leq x < 1 \\ x-1 & 1 \leq x < 2 \\ 0 & \text{otherwise} ...
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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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Constructing inversion method from a given pdf by finding inverse of cdf

The p.d.f. of the random variable $X$ is given by $f(x) = \begin{cases} e^{x-2} & \mbox{for $0 \leq x \leq 2$}, \\ e^{-x} & \mbox{for $x > 2$}, \\ 0 & \mbox{otherwise,} \end{cases}...
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How to combine two conditional CDFs

I am trying to reason about the following scenario: Let us have three random variables: $X$, $Y$, $Z$. $Y$ is independent of $Z$. Let us also have the following CDF's: $$F_X, F_{X \mid Y}, F_{X \mid Z}...
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Multivariate t-distribution CDF in python

Neither scipy nor statsmodels have multivariate t-distribution implemented. However, there are several code samples which implement Multivariate t-distribution PDF function, but I haven't seen any ...
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same cdf equals same expectation?

So, if $X$ and $Y$ are both continuous random variables with the same cdf, does that mean that their expectations are the same? And the same thing in case $X$ and $Y$ are both discrete. Thanks in ...
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R function to give me $P(X<x)$

I have this empirical discrete distribution with the respective percentiles (picture below). I want to know a R function which gives me $P(X\lt0.58)$ instead of $P(X\le 0.58)$ (given by ECDF function)....
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Expected value of a random variable by integrating $1-CDF$ when lower limit $a\neq 0$?

I have found several past answers on stack exchange (Find expected value using CDF) which explains why the expected value of a random variable as such: $$ E(X)=\int_{0}^{\infty}(1−F_X(x))\,\mathrm dx $...
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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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PDF of a log-normally distributed variable after tangens hyperbolicus transformation

Assume a variable $x_0>0$ with log-normally distributed noise, such that the observation $x$ of $x_0$ has the following PDF: $$ p(x\mid x_0) = \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{\left(\ln{(\...
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Finding $P(X>1)$ and $P(X\geq 1)$ from CDF

Introduction Statistics questions. I hope my question isn't too basic for this platform. I am given the CDF I've found $P(X \leq 1)$. I've found $P(X > 1)$ by $1-P(X \leq 1)$. Now I'm asked for $P(...
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Why do we use parametric distributions instead of empirical distributions?

The probability density function (pdf) is the first derivative of the cumulative distribution (cdf) for a continuous random variable. I take it that this only applies to well-defined distributions ...
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How is this CDF graph being obtained?

I'm working on computer vision research and I'm reading a paper about a probabilistic object detector. This object detector estimates 5DOF bounding boxes (x, y, width, length, heading) with each ...
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Positive or negatively bounded CDFs [closed]

If $X\in\mathbb{R}^n$ is a continuous random variable whose cumulative distribution function is ordinarily $$F_X(x) = \int_{-\infty}^{\infty} f_X(x) dx $$ what is the meaning of $$F_X(x) = \int_{0}^{\...
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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 ...
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Example of cumulative distribution function and the empirical distribution function [closed]

A random of 100 rolls of the die. The outcomes 1, 2, 3, 4, 5, 6, occurred 13, 19, 10, 17, 14, 27 times, respectively. Calculate the cumulative distribution function and the empirical distribution ...
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How to test if numerical function describes a valid probability distribution?

Suppose I can query a function $f$, but I don't have its closed form. We know the following things about $f$: $f(x) \geq 0$ for all $x$ $f$ is continuous Additionally, I can choose whether $f(x) \leq ...
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integrate multi-variate KDE over intervals to approximate probabilities in scipy

Background is I would like to work out the probabilities of certain events occurring to do this, Say I have three intervals: First (-inf, -1) Second ...
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Inverse transform sampling - CDF is not invertible

Suppose the cumulative distribution function $F$ is given but not invertible to use the inverse transform sampling technique (to compute $X=F^{-1}(Y)$). Do we have other alternative methods? I would ...
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What does i.i.d. mean for multivariate case?

When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
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Is The Jacobian Needed to Find CDF for R in Polar Coordinates?

I'm attempting to use inversion sampling to generate points on a disk according to the following PDF: $$ f(r) = \dfrac{2}{\pi(1+r^2)} $$ Here, the polar angle would just be a uniform random variable ...
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Artifact in CDF with real data?

Looking for some help explaining why intuition is failing me in exploring this data. I've binned by dataset by a predictor variable to examine the response variable through CDF plots. I realize it may ...
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Upper and lower limit of PDF

I have this example of a probability density function that is centered around 0. I would like to find out what are the lower and the upper limit of 95% of the data that is around 0. The goal is to ...
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Do all random variables' probability distributions have entropy?

Entropy of probability distributions is the weighted average of the log probabilities of each observation of a random variable. Does this mean that every random variable that has a probability ...

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