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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) $F(x)$...
1 vote
1 answer
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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, ...
Ishan Kashyap Hazarika's user avatar
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Distribution of a product of random variables

I have two independent distributions $X$ and $Y$. $X$ is defined by the piecewise CDF $$F_X(x) = \begin{cases} F_X^1(x) & x \in (-\infty, a_1)\\ F_X^2(x) & x \in [a_1, a_2)\\ F_X^3(x) & x \...
rkim's user avatar
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Is $F_X(t) > F_Y(t)$ a sufficient and necessary condition of $\mathbb{P}(X < Y) > 0.5$

I asked the question when conducting Mann–Whitney U test using the scipy implementation. The null hypothesis for one of the one-sided tests is $F_X(t) > F_Y(t)$, where $F$ denotes a CDF. I wonder ...
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Right continuity of cdf

Before asking, I want to let you know that I realize already there are different proofs for the right continuity of the cdf, however I would like to know if my proof of this is correct, as I assume it ...
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Question about CDFs

Suppose that I have two arbitrary random variables $Y,X$ with strictly increasing CDF. Let $\omega(X)$ be a known function of $X$ that is known in advance and $\mu$ be a scalar parameter. Assume that ...
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What makes a statistic valid for monte carlo simulation?

A while back I was reading Garland et. al. (1993) about studying whether two groups of animals, say herbivores and carnivores differ in their mean value for some trait, like the amount of territory ...
A Friendly Fish's user avatar
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How to understand intuitively the CDF formula for the maximum statistic of three iid rv’s? [duplicate]

Given that all three iid rv’s ($X_1, X_2, X_3$) have CDF $F(x)$, the formula for the CDF $G(y)$ of the largest rv ($Y=X_i$) among the three is: $G(y)=P(X_1 \leq y) \cdot P(X_2 \leq y) \cdot P(X_3 \leq ...
Michelle Zhuang's user avatar
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1 answer
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Derivation of a dynamical Generalized Pareto distribution

I'm currently reading a paper for my master thesis on the tail index estimation for asset returns using the peak over threshold method. In this paper the authors introduce the cumulative distribution ...
data_science_101's user avatar
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2 answers
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How to write a function for the normal copula in R?

How can I write the following function for the normal copula in R? $$ C_\theta(u, v)=\Phi_\theta\left(\Phi^{-1}(u), \Phi^{-1}(v)\right), $$ where $\Phi$ is the $N(0,1)$ cdf, $\Phi^{-1}$ is the ...
Aria's user avatar
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Derivative of the multivariate normal cumulative distribution function (CDF) with reparameterisation [duplicate]

I would like to learn how to calculate the derivatives of a multivariate normal cumulative distribution function (MVN CDF) w.r.t. certain elements by using the derivatives of the same MVN CDF w.r.t. ...
Kirin G.'s user avatar
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Is this the CDF of a known probability distribution?

Consider the following cumulative distribution function over $\mathbb{R}^{+}$ $F\left(x\right)\;=\;1+\mathcal{W}\left(-e^{-1-x}\right)$ where $\mathcal{W}\left(\cdot\right)$ is the Lambert W function. ...
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Does the negative multinomial distribution have a defined CDF?

I have a process that receives categorically distributed random inputs of 5 different types. The 5th type is considered "bad". If the process receives 4 bad inputs before receiving at least ...
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1 answer
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If $P(X\leq x') = a$ and $P(X\leq x'') = b$ where $b>a$. Is it always true that $x'' >x'$?

Let $X$ be a random variable. If $P(X\leq x') = a$ and $P(X\leq x'') = b$ where $b>a$. Is it always true that $x'' >x'$?
Alice's user avatar
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Question on the proof step in the theorem 1 of the Gap statistic paper

From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422), $\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
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Why is pnorm giving different results with X, sd and mean vs corresponding z value?

I have to calculate $P(X < 6)$ for $X \sim N(7,5)$. When I entered pnorm(6, mean=7, sd=5) into R, it gave me a result of <...
Peter Ambos's user avatar
2 votes
1 answer
49 views

linear Combination of Normal and T-Distributions

Consider the following probability distribution function (PDF): \begin{equation} p(x) = a\mathcal{N}(x; \mu, \sigma^2) + b \mathcal{T}(x; \mu, \tau^2, v) \; \; st. \; \;a + b = 1 \end{equation} $p(x)$ ...
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Comparing truncated distributions based on mean and cdf

Let $\tilde{x}$ and $\tilde{y}$ be random variables with pdfs $f_x(x)$ and $f_y(y)$ and cdfs $F_x(x)$ and $F_y(y)$. Given that $E[\tilde{x}] \geq E[\tilde{y}]$ $F_y(c) \geq F_x(c)$ for all $c \in \...
cat123's user avatar
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How do you test if two discrete ECDFs are drawn from the same population?

Background I have two Empirical Cumulative Distribution Functions (ECDFs) based on two samples of very different sizes. Sample 1: 1020 data points, Power-Law-like distribution, discrete data in the ...
Connor's user avatar
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What is the dagum type 2 probability distribution function?

I have been searching for dagum type 2 probability distribution function for several hours but all I have found is the cumulative density function of the mentioned distribution which is as follows: $$...
Sepideh Abadpour's user avatar
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Distribution of the number of iterations to achieve success

Let $Z=X+jY$ ($j$ is the imaginary unit), with $X\sim\mathcal{N}(\mu,1)$ and $Y\sim\mathcal{N}(0,1)$. I'm running an algorithm that at every iteration $k$ samples a complex number $z_k$ that follows $...
mateusgl's user avatar
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What is the PMF of the Zero-Truncated Skellam distribution?

Vaguely related to the notion of a Lindley equation, I am considering a recurrence $$L_{t+1} := L_t + \max \left( 0, A_t - S_t \right)$$ where $$L_t$$ is the number of customers in a queue system at ...
Galen's user avatar
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2 votes
1 answer
273 views

Is the cumulative distribution function of a r.v. X strictly increasing (X -) almost everywhere?

Let $X$ be a random variable and $F_X(x) = P(X \le x)$ its cumulative distribution function (cdf). $P_X$ is the probability measure induced by $X$, which is defined by $P_X((a,b)) = P(X^{-1}((a,b))$ ...
ChrisL's user avatar
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Confidence interval for sum of product of scaled binomial random variables

I have discrete, independent, but not necessarily identically distributed random variables $X_1,\dots,X_n$ that take on non-negative integer values. Each random variable has unknown distribution ...
Efficiency's user avatar
1 vote
1 answer
48 views

Transforming data with a fitted distribution function

I have a bivariate dataset on $[0,1]^2$ in which I am interested in fitting a joint distribution. I fit a Gaussian copula but am unsure how to judge if it's a good fit. I tried transforming my data ...
Bpe's user avatar
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3 answers
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Problem with Cumulative distribution function

I can't understand this cumulative distribution function. I would like to calculate the data distribution function: ...
Zollikofen4's user avatar
3 votes
1 answer
274 views

Correspondence between the "density function of a probability measure" and the "probability density function" (PDF)

Question. If there is a one to one correspondence between a "borel probability measure" on the line $\mathbb{R}$ and a "cumulative distribution function" (CDF) (please see on page ...
Ommo's user avatar
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1 answer
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Why P(X=x) is not zero here?

From the CDF,I feel that the random variable here is continuous.So shouldn't the P(X=x) equals 0 here and we only need to find P(X>=1).
Aman's user avatar
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2 votes
1 answer
108 views

Quantile function of multivariate distributions from empirical samples

Let's say I have a k-dimensional multivariate normal distribution $MVN(0,\Sigma)$. Denote random vector $X \sim MVN(0,\Sigma)$ as $X = (x_1, x_2, \dots, x_k)$. It is trivial to find $P(x_1 \leq c_1, ...
David Wang's user avatar
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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 ...
TyDurden's user avatar
3 votes
0 answers
48 views

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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4 votes
2 answers
114 views

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:...
involuptory's user avatar
1 vote
0 answers
58 views

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 ...
borcherds's user avatar
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0 answers
15 views

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, ...
johnsmith's user avatar
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1 vote
1 answer
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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 ...
strateeg32's user avatar
4 votes
1 answer
126 views

How to find distribution of $\Phi \left( x \right)$

Given $X \sim N\left(\mu, \sigma^2 \right)$, I want to find the distribution of $$Y = \Phi \left( X \right),$$ with $\Phi \left(x\right) = \int_{-\infty}^{x} \frac{1}{ \sqrt{2\pi}} e^{-\frac{t^2}{2}}...
augustine's user avatar
1 vote
1 answer
137 views

Calculating PDF conditioned on event

I'm confused about problems where we calculate a PDF conditioned on an event. Consider this simple problem: We have two random variables, X and Y, X is uniformly distributed on [a,b], and Y is uniform ...
MohammadAli Zeraatkar's user avatar
1 vote
0 answers
84 views

How does a fast Fourier transform yield the probabilities for a multinomial random variable?

Let $(A,B,C)$ follow a multinomial distribution: $$(A,B,C) \sim \text{MultiNom}\left(n=100,p_1=p_2=p_3=\frac13\right).$$ Define $X$, a discrete random variable as $$X = f(A,C) = 2A + 3B + 4C = 2A + 3(...
Gregg H's user avatar
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Approximate distribution of random variable similar to studentized mean R.V?

It is well known that the distribution of the studentized mean, i.e., $T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} $, can be approximated (...
Jaimin Shah's user avatar
1 vote
1 answer
34 views

Problem with deriving the cumulative distribution from the density function

Consider the continuous distribution with density function $$ p(x) = \frac{1}{2}\cos(x) \;, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}. $$ I want to derive the cumulative distribution function for ...
dedeCe's user avatar
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2 votes
1 answer
116 views

Fitting a nonlinear model for a CDF

I have two questions in general here. Suppose I am recording data in time and the response that I am collecting is a monotonic curve that goes from 0 to 1 (sort of a like a CDF). I was thinking of ...
John Smith's user avatar
0 votes
1 answer
117 views

Compare CCDF of datasets with different sample size

I have 3 dataframes with the same structure (each dataframe includes a different type of tweet). Here are the columns of dataframes: id, ...
mOna's user avatar
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1 vote
0 answers
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Is there a package/function in R, Matlab or Python to perform smooth non-parametric fitting of empirical CDF data? [closed]

I have numerical process that generates percentiles values of a certain random variable $X$, meaning that I have $x_i$ values and their corresponding probability values $\hat{F}(x_i)$. I want to fit a ...
Xavier Romão's user avatar
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0 answers
95 views

Approximation on Inverse Mills ratio for the normal R.V

I've come across several approximations for Mills ratio, but I haven't found any good ones for the Inverse Mills ratio. Is there any known closed-form approximation for the Inverse Mills ratio (link) ...
Jaimin Shah's user avatar
3 votes
1 answer
202 views

Distribution of sum of a discrete uniform and a uniform on (0,2)

Let $U$ be uniformly distributed on the interval $(0, 2)$ and let $V$ be an independent random variable which has a discrete uniform distribution on ${0, 1, . . . , n}$. i.e. $P\{V = i\} =\frac{1}{n + ...
Tapi's user avatar
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0 answers
27 views

Question about joint cdf

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 \...
Parinn's user avatar
  • 83
2 votes
1 answer
85 views

CDF of $\max$ under conditions

Let, \begin{equation} g(\alpha,\beta) = \begin{cases} \frac{\alpha}{\beta}, & \text{if } \alpha > \beta \\ 0, & \text{if } \alpha \leq \beta \end{cases} \end{equation} I want to find ...
Frey's user avatar
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1 answer
102 views

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. ...
Pedro Rodrigues's user avatar
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0 answers
61 views

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 ...
smv's user avatar
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1 vote
0 answers
25 views

Fit generalized linear mixed model (with lme4 or other) to cumulative data of a continuous variable

I have measurements of resin production of pine, which are taking tapping the tree, that is, making a physical wound and collecting the resin in a pot. When the pot is full we replace it with an empty ...
Asier's user avatar
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