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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Approximating non-integer median for CDF of discrete variable

In my googling, it seems the proper way to find the median of a cdf of a discrete variable is to stick to the discrete values provided, even if you overshoot and end up with an x where P(X <= x) &...
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Proving sufficiency of properties of pdf/pmf [closed]

We have the following necessary and sufficient conditions for a pdf/pmf $f_X(x)$: $f_X(x)\geq 0$ for all $x$ $\sum_x f_X(x)=1$ if pmf or $\int_{-\infty}^\infty f_X(x)dx=1$ if pdf To prove necessity ...
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Inverse of cumulative density function for Multivariate Normal Distribution

How do I calculate the inverse of the cumulative distribution function (CDF) of a multivariate normal distribution? Does it even exist for the multivariate case? I know this is possible for a ...
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Which distribution creates an exponential-like pattern in x log scale?

I've been reading this very nice paper by Baltrunas et al. and I would like to have a distribution that looks as much as possible like the empirical data the authors found in the figure below: I don'...
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Out-of-Specification for log-normal data

I want to calculate the Out-Of-Specification (OOS) probability of my dataset (probability of having a result which is greater than the acceptance limit). I have noticed that my results follow a log-...
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Does dependency imply an equation?

On regression, we usually think of dependency in terms of an equation relation between variables. For instance, we think that $Y$ "depends" on $X$ If $$E[Y|X] = g(X) + \epsilon \quad \mbox{...
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How can this CDF be decreasing? (power-law)

I am reading the first vignette of the powerRlaw R package. It uses the moby dataset, which is an array of word frequency. Each element is a word, and the value if ...
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How to prove stochastic dominance of negative binomial random variable

Let $X\sim NegBin(n_1,p)$ and $Y\sim NegBin(n_2,p)$ with $n_1>n_2$. How do I show that $X$ stochastically dominates $Y$ (i.e. $F_X(x)\le F_Y(x)\quad\forall\:x\in\Bbb R$)? My try Since a Negative ...
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Generating an analytical copula for an example problem

I am currently doing research that requires me to understand dependence modeling. As a first step, I am reading An Introduction to Copulas. I am, stuck on the first example problem which I have re-...
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A proof that the median is a nonlinear statistical functional

This question is with reference to the top answer (by @StephanKolassa) to this question. Let $F$ and $G$ be CDFs and define $$H(x)=aF(x)+(1-a)G(x)$$ with $a\in [0, 1]$. Now suppose $F$ and $G$ are ...
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Clarification regarding comonotonic random variables

According to wikipedia, an equivalent definition of a comonotonic random vector is: A $\mathbb{R}^n$-valued vector is comonotonic iff it agrees in distribution with a random vector where all ...
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Standard Errors of ECDF with Potential Model Fit Error

I have data and a proposed likelihood. I used MLE to fit the likelihood, but it is a very complicated likelihood and there is some estimation error of the model. The likelihood may be correct, but ...
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How to compute quantile of a mixed distribution? [duplicate]

A mixed distribution where cumulative probability distribution function (CDF) is given by G(x)= (1-p)H(x)+pF(x) where, p=0.2 (assumed in this case as it ranges ...
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Memoryless property: how is $F^c(2) = F^c(1)F^c(1) = (F^c(1))^2$ and $F^c(1/2) = (F^c(1))^{1/2}$ implied?

I am currently studying the textbook Modeling and analysis of stochastic systems, third edition, by Kulkarni. Chapter 5.1.1 Memoryless Property says the following: We begin with the definition of the ...
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Memoryless property: how does $P(X > s + t \mid X > s) = P(X > t), s, t \ge 0$ imply that $F^c(s + t) = F^c(s) F^c(t), s, t \ge 0$? [closed]

I am currently studying the textbook Modeling and analysis of stochastic systems, third edition, by Kulkarni. Chapter 5.1.1 Memoryless Property says the following: We begin with the definition of the ...
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Binomial-CDF/binomial-test for classifier significance testing

General Problem Description and Goal I set up a matching algorithm that matches a user input (string) with a list of possible values (words) that is conclusive but very large (some 5 digit ...
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Calculating a variation of the inverse mills ratio in R

The equation below, related to the inverse mill ratio, comes from Chiburis and Lokshin (2007), page 3 (169 of the journal). (Free Access). I want to calculate this formula in R, based on output from a ...
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Picking a specific estimated CDF from a set of CDFs provided by an ECDF

Let $F_X$ be a CDF of an unknown random variable $X$. If we have independent samples $x_1, x_2, \ldots, x_n$ of $X$ then we can estimate $F_X$ non-parametrically using an ECDF $\hat{F}_n$. By Central ...
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Notation for ECDF

I'm reviewing statistical functionals and U-statistics, trying to make notes, and I am tripping on notation. From my understanding, $X$ is used to denote a random variable and $x$ is used to denote an ...
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Cumulative Distribution Function of $S_{N_{t}}$ where $S_{N_{t}}$ is the time of the last arrival in $[0, t]$

I am confused on this problem. My professor gave this as the solution: $S_{N_{T}}$ is the time of the last arrival in $[0, t]$. For $0 < x \leq t, P(S_{N_{T}} \leq x) \sum_{k=0}^{\infty} P(S_{N_{T}}...
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Enforcing conditions on truncated exponential distribution

The CDF for an exponential distribution of rate $\lambda$ truncated at T is $F(t) = \frac{1-e^{-\lambda t}}{1-e^{-\lambda T}}$. (for $t<T$, else 0). I would like to determine $\lambda$ and $T$ such ...
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How do I calculate the cumulative probability of an observation fitted with a mixed effects model?

I am interested in getting the cumulative probability of an observation that is fitted with a generalized linear mixed effects model. For a linear model with gaussian distribution this seems simple. (...
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Gaussian Distribution: How to calculate the Cumulative Distribution Formula (CDF) from the Probability Density Function (PDF)? + Error Function? [duplicate]

I understand that we can calculate the probability density function (PDF) by computing the derivative of the cumulative distribution formula (CDF), since the CDF is the antiderivative of the PDF. I ...
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CDF for local time of non-standard Brownian Motion

$L_B(1,0)$ is the local time variable for the standard Brownian motion $B(t)$, and its cumulative distribution function is $P(L_B(1,0) \leq x)=2\Phi(x)-1$ for $x \geq 0$ and $0$ otherwise, in which $\...
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Comparing CDFs, Kolmogorov-Smirnov, Max/Average?

I'm in the process of building a python group of classes that will calculate many CDFs for a large number of samples. I then want to compare the CDFs of all the samples to see if any two samples could ...
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Joint cumulative distribution

I'm studying joint distributions of two random variables, $X$ and $Y$. Ross's book (Chapter 6) defines the joint CDF as $F(a, b) = \mathbb P(X \le a, Y \le b)$ and the PDF as $f(a,b) = \frac{\partial}{...
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Calculate probabilites of intersections and union of two i.i.d. Exp CDF

Lets say I have two Exponential $X$ and $Y$ independent variables distributed $Exp(1)$ The CDF $F(x)=P(X \leq x)$ is given by $1-e^{-x}$ The Survival function $1-F(x) = S(x)=P(X > x)$ is given by $...
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Suppose $X_n$ is a record value. Will $F(X_n)$ be a record value as well?

Let $X_1,\cdots,X_n$ be independent and identically distributed random variables with distribution $F$. We say that a record occurs at time $n,n>0$ and has values $X_n$ if $X_n > \max (X_1,\...
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Drug Efficacy - Probability Plot for Sputnik V

On the Wikipedia page of Sputnik V vaccine, results of Phase III study are reported, where the drug efficacy is visualized for the first and the second vaccination (attached picture). Does this plot ...
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Cumulative distribution function of normal distribution as a series

According to Wikipedia the cumulative distribution function of the standard normal distribution can be approximated with the ...
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How do I use CDF function of a stochastic demand to express expected profit of a firm

Could you please help with transforming the equation (1) of the expected profit of manufacture to equation (2), which using the CDF to show the expected profit? r is the profit, D is the demand of the ...
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How do I interpret results of Cumulative Link Mixed Model? Ordinal dependent variable

I have data that looking at disease ratings of plant roots on a scale from (0-5). Five being the most severe and 0 meaning no disease. I have 10 fungicide seed treatments including a control. ...
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Can we say anything about the limit of the Mill's Ratio for the Normal Distribution?

Is there anything we can say about $\lim_{x\rightarrow \infty}\frac{F(x)}{f(x)}$ where $F(x)$ is the CDF of the normal distribution and $f(x)$ is the pdf? I know this function is increasing, but what ...
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Deriving the PDF/CDF of Slash distribution

The Slash distribution is defined as $\frac{Z}{U}$ when $Z\sim N(0,1)$ and $U\sim U(0,1)$. I wonder how one reach's the CDF/PDF of this distribution?
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R: Left-censoring a probablity distribution function

I would like to left-censor (at zero) a probablity distribution function, but I just can't find a way to implement this in R. I have reviewed previous questions about censoring, but none have provided ...
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Intuitively understanding $F_X(X)$ (r.v. as argument) [duplicate]

When considering some cdf $F_X(x)$ — e.g. from here — I’m having a hard time trying to understand what $F_X(X)$ really means. Expanding gives $P(X \leq X)$, which at first glance should always equal $...
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RV's with same mean/variance satisying Ohlin's lemma

I am trying to find two random variables X,Y with same mean and variance such that Ohlin's Lemma holds. That is, there exists some $x_0$ such that $F_X(x) \leq F_Y(x)$ for $x < x_0$ and $F_X(x) \...
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Simulate CDF curve for penetration/adoption extrapolation

I'd like to be able to plot a line like the cumulative distribution function for the normal distribution, because it's useful for simulating the adoption curve: Specifically, I'd like to be able to ...
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How to write down a cumulative distribution function that consists of two distributions

I am being asked to write a CDF of a random variable $X$. I know that there is $0.5$ probability that $X=5$ and $0.5$ probability that $X$ follows the exponential distribution with parameter $7$. I ...
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Problem based on Cumulative distribution function

I have been given the following cumulative distribution function. I need to compute the probability $P(1\leq X<2)$: $$F(x)=\begin{cases} 0 & \text{ if } x<0 \\[1ex] \frac{x^2}{10} & \...
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Is $Z_t = U\sin(2\pi t) + V\cos(2\pi t)$ is strictly stationary?

Let $Z_t = U\sin(2\pi t) + V\cos(2\pi t)$, where $U$ and $V$ are independent random variables each with mean 0 and variance 1. Is $Z_t$ strictly stationary? Answer: I have proven that $Z_t$ is ...
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What information is required to aggregate percentiles

I have several thousand large datasets that are too big to fit into memory at once, so I need to keep them separate. It is easy enough to get the count, mean, std dev, min and max for the whole ...
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Finding the CDF of a a sequence of independent random variables

Let $Z_t$, where $t$ is even, be a sequence of independent random variables defined as, $$Z_t = \left\{\begin{array}{ccc} +1 & , & p = \frac{1}{2} \\ -1 & , & p = \frac{1}{2}\end{array}...
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how to generate data from a distribution whose cdf is not in closed form? [duplicate]

I am working on a distribution whose pdf and cdf is $$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
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how to generate data from cdf which is not in closed form?

i am working on a distribution whose pdf and cdf is $$f(x,\alpha,\beta)=\frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta}}{(1+(\frac{x}{\alpha})^{\beta})^{2}}\frac{\sin(\frac{\pi}{\beta})}{(\frac{...
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How to Solve For the Inverse Cumulative Distribution Function of a Double-Exponential Probability Density Function

I'm stuck on figuring out how to sample data from a fake/known double-exponential PDF for a lab project involving C. elegans egg-laying rate data. I need help with figuring out if there's an exact ...
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The formula for conditional expectation in terms of joint cdf

We know that covariance can be written as a function of marginals and joint CDFs, namely $$\newcommand{\cov}{\operatorname{cov}}\newcommand{\d}{\mathrm{d}}\cov(X,Y) = \iint (F_{X,Y}(x,y) - F_X(x)F_Y(y)...
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Constructing portability density function by convolution of two PDFs and acquire ICDF

I need Gaussian-convoluted gamma distribution to fit my data, but the program I'm using doesn't allow construction of custom PDF. Let's call it pdf_g2. $$\mathrm{pdf_{g2}}(x,a,b,c,d) = conv[\mathrm{...
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Meaning of uniform percentile rank distribution?

I'm new to statistics and I've been following Think Stats 2. I've just gotten to Cumulative Distribution Functions. I have some questions regarding uniform distribution: What does it mean for the CDF ...

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