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.

Filter by
Sorted by
Tagged with
0
votes
1answer
64 views

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 $y = sin(x)$ ...
0
votes
3answers
124 views

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 $$...
4
votes
1answer
301 views

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 ...
-1
votes
0answers
17 views

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 $T_n = nY_n$, ...
0
votes
1answer
126 views

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} ...
1
vote
0answers
48 views

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, ...
2
votes
1answer
123 views

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}...
1
vote
0answers
63 views

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}...
0
votes
0answers
81 views

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 ...
1
vote
1answer
90 views

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 ...
3
votes
1answer
82 views

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)....
4
votes
3answers
408 views

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 $...
2
votes
1answer
62 views

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 ...
1
vote
1answer
53 views

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 ...
3
votes
1answer
67 views

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{(\...
0
votes
1answer
92 views

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(...
7
votes
1answer
490 views

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 ...
0
votes
0answers
43 views

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 ...
-3
votes
1answer
62 views

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}^{\...
0
votes
0answers
22 views

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 ...
2
votes
0answers
124 views

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 $S_1$, ...
-1
votes
1answer
211 views

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 $$1-F(x)$$
2
votes
3answers
2k views

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*} ...
0
votes
0answers
123 views

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 ...
0
votes
1answer
153 views

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 ...
0
votes
0answers
21 views

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 ...
0
votes
0answers
86 views

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 ...
10
votes
3answers
1k views

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 ...
3
votes
2answers
206 views

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 ...
0
votes
1answer
47 views

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 ...
0
votes
0answers
103 views

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 ...
1
vote
2answers
152 views

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 ...
2
votes
1answer
102 views

Which financial time series have a PDF and/or a CDF? [closed]

Consider the following types of financial time series for a single publicly-listed stock: Price data Log returns Cumulative returns Each is computed from the item listed before it: log returns are ...
2
votes
1answer
144 views

Why copula based on CDF instead of PDF

I do understand the mathematic behind probability density function( PDF) and cumulative distribution function (CDF). My problem starts when I try to understand why copula relies on CDF and not on PDF. ...
0
votes
1answer
240 views

How can i find empirical survival function using survival function in R?

The survival function is given by: S(y; α, λ) = (α/α−1)* (1 − α^(−e^(−λy ))), if α is not equal 1 = e^(−λy) if α =1 y = 1 4 4 7 11 13 15 15 17 18 19 19 20 20 ...
0
votes
1answer
97 views

How to find ks test statistic using the given maximum likelihood estimator values and a sorted data in R?

The random variable Y is said to have a two-parameter APE distribution denoted by APE(α, λ), with the shape and scale parameters as α > 0 and λ > 0, respectively, if the PDF of Y for y > 0 is ...
1
vote
0answers
24 views

Does moments minimization implies CDF maximization?

I have a non-negative random variable $X$ with a continuously differentiable PDF $f(x|\theta)$, where $x\geq0$ and $\theta$ is the distribution parameter. The corresponding CDF is $F(x|\theta)$. I don'...
1
vote
1answer
207 views

calculating CDF of kth order statistic

I have recently started probability and statistics on my own. Pls help in understanding below. $x_{(k)}$ is kth smallest random variable from sample of n iids ($x_1 \to x_n$) For calculating CDF of ...
1
vote
0answers
35 views

Find $x_0$ that satifies $\mathbb{P}(X \leq x_0) = 0.75$

Suppose that $X$ is a continuous random variable with probability density function: $$\begin{cases} x & 0 \leq x < 1, \\[6pt] 2-x & 1 \leq x < 2, \\[6pt] 0 & \text{otherwise}. \\...
0
votes
0answers
120 views

Finding PMF, CDF of a piecewise function of an RV

Here's the question: Let $Z$ have CDF $F$ and pdf $f$ and let $A$ be a subset of the real line. Further, let \begin{cases} W = 1 & \text{if $Z \in A$} \\ ...
1
vote
2answers
112 views

How do I find the PDF from a multidimensional CDF with indicator functions?

I have what I'm sure is a very stupid question. When I have a two-dimensional random variable $\tilde{X}=(X_1,X_2)$ with the cdf $F(x_1,x_2)=(kx_1^2I_{(0,1)}(x_1)+I_{[1,\infty)}(x_1))(kx_2^2I_{(0,1)}(...
0
votes
1answer
55 views

difference in CDFs and pdfs of joint distribution of two random variables

We know that the joint probability function of two independent random variables is just the product of their respective pdfs. On the same lines, .can we say that if we multiply the cumulative density ...
0
votes
1answer
103 views

Deriving Inverse of Cumulative Distribution function [closed]

let $f(x)$ be a probability distribution, and $g(x)$ be the cumulative distribution of that probability distribution (CDF) By definition, CDF is: $ g(x)=\int_{-\infty}^{x} f(x) dx $ Given $g(x)$, I ...
36
votes
4answers
3k views

Intuitive explanation of Kolmogorov Smirnov Test

What is the cleanest, easiest way to explain someone the concept of Kolmogorov Smirnov Test? What does it intuitively mean? It's a concept that I have difficulty in articulating - especially when ...
0
votes
1answer
138 views

An example of continuous random variable X > 0 with finite second moment but Infinite third moment [duplicate]

Can someone construct an example of this? i.e., $E[X^2] < \infty$ but $E[X^3] = \infty$. Results could be in terms of pdf, or cdf, or survival function. Justification would be appreciated
1
vote
1answer
372 views

How to decompose a CDF into discrete and continuous parts?

I understand that any C.D.F may be represented in the form $$F(x) = p_1F^d(x) + p_2F^c(x)\,,$$ where $F^d(x)$ represents discrete c.d.f , $F^c(x)$ represents continuous c.d.f and $p_1+ p_2=1$. ...
0
votes
0answers
42 views

Goodness of fit that puts high weight towards the tail of the distributions

I have two distributions A and B and I am looking for a goodness of fit test that measures how much the tail of A matches (or fail to match) the tail of B. Alternatively, I am looking for a test that ...
1
vote
0answers
48 views

Generate random variates and cdf from Tweedie distribution

I have to generate some random variates from a Tweedie distribution with the following parameters: $p = 1$ $\mu = \bar{\mu}$ $\phi = \bar{\phi}$ I do not have direct access to a function that is able ...
1
vote
0answers
36 views

Tweedie cumulative distribution function [closed]

I am trying to calculate the cdf of a Tweedie distribution. The R function ptweedie gives me some weird results and I do not understand why. Here some code in R <...
0
votes
1answer
50 views

Why are there two ways to write PDF and CDF functions?

I often see PDF and CDF functions written as either $f_X(x)$ or $f(x)$ for PDF or $F_X(x)$ or $F(x)$ for CDF. In what situations would you use either notation? Like what is the point of having ...

1 2
3
4 5
14