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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Asymptotic Distribution of Minimum Uniform Random Variables

I've been working on this problem for a while, and I've made some progress, but I'm still stuck on some parts. I was hoping to get some assistance with this! Let $M_n = \min(X_1, ..., X_n)$ where $...
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CDF for a probit model

The cdf for a probit model is: $$ \Phi(\varepsilon)=\int_{-\infty}^\varepsilon \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{t^{2}}{2}\right) \, dt $$ My very simply question—that I should know the answer ...
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Fitting a normal CDF using proportion data

I have the following data (prop is like empirical CDF): ...
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Mix pdf and cdf in binary response model [duplicate]

Let's suppose that I have a model that tells me how likely is for an event to have come after a certain time lapsed, given by some kind an exponential distribution, i.e. $$ \mathbb{P}(T_E < t) = \...
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49 views

Inverse-normal CDF approximation in Excel, Python or R

I read that the implementations of Inverse-normal cumulative distribution function (CDF) /quantile / ppf in R, Python (scipy) and Excel give similar results. However, I can't find the very formulae ...
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Number of times an object gets sampled when sampling $N$ times with replacement from $M$ objects [closed]

Take $N$ draws with replacement from $M$ unique objects. As $N$ goes up, what is the distribution over the increment to the number of times I get each $M$? So say $M$ is 3 and $N$ is one. The ...
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R Multivariate Normal CDF

I am trying to estimate the CDF of a MVN distribution for some given ranges ([-Inf,mu], [mu,mu+3], [mu+3,Inf]). In R using package mvtnorm and function pmvnorm, with some dummy data: ...
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How to calculate the cumulative distribution function of a GEV distribution when $1+\xi(x-\mu)/\sigma\le0$?

I don't have a stats background let alone one in extreme value theory, and I have what I imagine is a simple question but one I that haven't been able to find the answer to. The cumulative ...
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CDF for f(x) = 0.5e^-|x|

This is the full question: "If a random variable has density f(x)= 0.5e^-|x|, for x∈R, find the cumulative distribution function". I know that to find cdf from the pdf you would have to integrate ...
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Basic CDF / CDF Notation Question

The following is copied from some old course notes: $F_{X-\theta}(x)=P(X-\theta < x) = P(X<x+\theta)=F_\theta(x+\theta)=F_0(x+\theta-\theta)=F_0(x)$ Is anyone able to explain the $P(X<x+\...
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What is the difference of $\Sigma$ esimation of Gaussian Copula based on known CDFs VS unknowns

Recently, I read this web page which explains the Copula package in R. A question occurred to me. Consider a data set $D_{n\times d}$ which $n$ is the number of samples and $d$ is the number of RVs. ...
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What information does a probability density function (PDF) graph provide?

This sounds like a simple question and I know PDF graphs are used a lot in presentations and financial publications. Yet, what information does it actually provide? The CDF actually gives you ...
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Calculating CDF for Dry Days (0mm/day)?

I am working on daily precipitation data and need to calculate cumulative Distribution Function (CDF) of daily precipitation data, however, I don’t understand how to transform 0 mm/day (dry days) to ...
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Convergence in law of the remedian

I try to understand the theroem 2 of this article about the remedian (https://pdfs.semanticscholar.org/3d64/5e60691838bf4699e79458d96930ba7bf24e.pdf) I will try to phrase it more general so that you ...
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How do I get the CDF of a gamma distribution with mean and sd?

I have the mean and standard deviation of my data, which I determined follows a gamma distribution. I don't understand the function I found online for the CDF of a gamma distribution because of the ...
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Analytical solution to the multivariate CDF given multivariate pdf

Is there any way of approximating or analytically solving the below CDF (let's say even for $n\to\infty$)? I am trying to find the below probability: \begin{align} &P\left[X_{2}-X_{1} \leq 0,...
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How to prove KL(q(z)|p(z)) = E_q(z) [ log f(z) ] where f is the CDF of p?

As title. It was used in https://arxiv.org/abs/1905.10549 without proving.
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If $X$ follows standard normal distribution, find the correlation coefficient between $X$ and $\Phi(X)$

If $X$ follows standard normal distribution, find the correlation coefficient between $X$ and $\Phi(X)$, where $\Phi(X)$ is the cdf of $X$. My attempt is: First we have to calculate $Cov(X, \Phi(X))$...
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Derivation of CDF of a function that results in an exponential distribution

I was looking through wiki's treatment on the title topic in https://en.wikipedia.org/wiki/Random_variable and am completely stumped on this particular section: There are several specifics that ...
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Can I increase the sample size by generating random numbers to apply the Chi-Square Goodness of Fit Test?

Does increasing the sample size by random number generation change the distribution? I have a sample of size 8. Each sample value represents the number of bus arrivals at a bus stop every 15 minutes. ...
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How to find quantiles and likelihoods of mixture distributions?

My PDF: M was estimated and found to be 5. I need to work out the quartiles for the PDF above. In addition, I need to use different methods of estimation to estimate the parameters. So far I've ...
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Tcdf and invT with incomplete beta [duplicate]

I am working in python. I have a function for incomplete beta. Iow would I calculate tCDF and invT using this? the incomplete beta function that I have is this: ...
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Joint CDF of M(t) and B(t), where B(t) is the standard BM and M(t) is maximum value of standard BM on [0,t]

We have to find - $F_{M(t),B(t)}(m,x) = P(M(t) \leq m, B(t) \leq x)$. $T_{m} = inf\{t\geq 0: B(t) = m\}$. We know that, $$ P(M(t) \geq m, B(t) \leq x) = P(T_{m} \leq t, B(t) \leq x)$$, $$ = P(T_{m}...
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Strategies for predicting values of a time-dependent CDF, given covariates

I've got data that looks like this: ...
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Consistent estimator and distribution function

a general question: If the distribution function $F_n$ of some estimator $T_n$ suffices \lim_{n \rightarrow \infty} F_n(x) = 1 \text{ or } 0 \forall x}. Does that imply that $T_n$ is consistent? I ...
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Calculate the derivative of the CDF with respect to the mean value [duplicate]

I want to derive the cumulative density function (cdf) for variables following normal distribution with respect to the parameters of the cdf (such as the mean or the standard deviation)
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Conditional transformation of variables

I've seen a trick for finding the p.d.f of $r(X,Y)$ where $X$ and $Y$ are r.v's by first calculating the cdf i.e $P(r(X,Y) \leq l)$ and then differentiating to find the pdf. So if $\Omega = \{(x,y) | ...
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Multiply CDF by constant, what is the expected value of this “new” CDF? [closed]

Specifically, I want to multiply $F_X(x)$ by $E(X)$, so I have $$ ??? = E(X)\cdot F_X(x) = \int^b_a xf_X(x)dx\cdot \int^b_a f_X(x)dx \overset{?}{=}\int^b_ax\Big(f_X(x)\Big)^2dx $$ Is there a way to ...
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What is the expected value of half a standard normal distribution?

You have a normal distribution with mean of 0 and variance of 1. Keeping the same probabilities and focusing only on half of the distribution (other half has it's original probabilities but x values ...
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can you help me to do a difference of CDF?

I have 2 CDF's with equal number of points that I want to compare. These are from: Temperature of 1 month from 2012 Mean temperature across months What can I do to obtain this difference, this is ...
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cumulative distribution functions (CDFs) [duplicate]

i want to know why is important use CDF for this analysis of TMY (typical meteorological year) because i have the data of the month for compare with the long term mean This the example of the manual ...
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How to find the conditional CDF based on observed data in R [closed]

If we have two samples (generally their distribution is not known),say $X\sim N(0,1)$, $Y|X\sim N(X,X^2/2)$. Can we recover the conditional CDF of $Y|X$ based on the observed samples in R? ...
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Approximate density from moments and quantiles, then sample from it

Situation I need to send R code to a third party to run estimations for me (I will not be able to work with the data directly). I want to simulate data to test some of the estimators before sending ...
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If $F_X(z) > F_Y (z)$ for all $z\in \mathbb{R}$ then $P(X < Y ) > 0$?

I came across this question in a review of an old exam I took. I didn't get the answer correctly then, and I'm struggling to figure the answer out now. Can anyone help me reason through this? ...
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Integration by parts for multivariable cumulative distribution function

How can I integrate by parts $$\int_A (y_1+\cdots+y_n) \,dF(y_1,...,y_n),$$ where $F$ is the cumulative distribution function for some random vector, $A$ is some Borel bounded set in ${\mathbb R}^n$.? ...
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Upper bound of normal cdf

Random variable $X\sim N(0,1)$. Show that, $P(X\geq c) \leq e^{-ct+ \frac{t^{2}}{2}}$ for $c>0$ and for all $t$ in $R$. I found that $P(X\geq c) = \Phi(-c)$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)...
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Mapping a range of values such that the resulting distribution is uniform [duplicate]

I have a set of values. Let's call the set X with values ... . Those values in [0, 1] have a non uniform distribution (empirically measured). I would like to re-map those values on [0, 1] such that ...
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An approximation to the cdf of the normal from a pdf?

In this paper (p. 36), authors wrote $$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$ Bellow we will use the approximation $$p(n,T) =...
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Simulation: Generate random numbers that cluster around an average? [closed]

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
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Kolmogorov–Smirnov test on text data

The Kolmogorov–Smirnov test a very efficient way to determine if two samples are significantly different from each other or whether the CDF between two different samples fit each other. This can be ...
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Efficiently Computing The Beta CDF [duplicate]

I am using numba to JIT compile some looped python functions as part of a larger application. Ideally, everything will run in numba's "no python" mode, such that the loop can be parallelised. One of ...
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Finding the joint CDF using the joint PDF; why can't I do this?

Find the joint CDF of the independent random variables $X$ and $Y$, where $f_X(x)=x/2, 0\le x \le 2, $ and $f_Y(y)=2y, 0 \le y \le 1$. To do this, we can find the CDF separately for each of the ...
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Compute $P(Y<3X)$ using joint PDF

I'm given a joint pdf $f_{X,Y}(x,y)=2e^{-x-y}, 0<x<y, 0<y $ and asked to compute $P(Y<3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this ...
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Tail behaviour of normal cdf?

Q: What is the tail behaviour of $\log \Phi(t)$ as $t \to \infty$? Since $\Phi(t) \to 1$ as $t \to \infty$, we know that $\log \Phi(t)\to 0$, but I would like to know at what rate this function ...
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Order Statistics; Finding the probability that the first sample is < 0.6, and the last sample is > 0.6

Here is the problem statement below: A random sample of size 5 is drawn from the pdf $f_Y(y)=2y, 0\le y \le1$. Calculate $P(Y_1^{'} < 0.6 < Y_5^{'})$. Here, using formulas for order ...
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Definition of CDF of discrete RV

In many different (serious and good) statistics books I find different definitions of CDF of a discrete RV. The difference is the equal sign at the index of the summation sign. The first is: $$F(x) = ...
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Convergence in Distribution, Argument Converging in Probability

Suppose $\lim_{n\to\infty}P(X_{n}\leq x) = P(X\leq x)$ and that $A_{n} \stackrel{p}{\longrightarrow} a$, where $a$ is a continuity point of $F_{X}(x) = P(X\leq x)$. Is it the case that $\lim_{n\to\...
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Finding the CDF given marginal PDF's; setting bounds

In this question, I'm having a hard time understanding how specifically to set the bounds for the CDF. Let $X$ and $Y$ be independent variables. Find the CDF of $W=Y/X$ using the marginal PDFs ...
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PDF transformation for y=|x|

Suppose I have the random variable X with a pdf: $$f(x)=exp(-(x+1)) u(x+1)$$ where u is the unit step function; such that u = 0 for x<-1 and u=1 for x>-1 $$y= |x|$$ for $$-1<x<1$$ ...
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How to relate beta CDF to student-t CDF? [duplicate]

We can relate the student-t and beta distributions as such: If $X$ has a Student's t-distribution with degree of freedom $\nu$ then one can obtain a Beta distribution: $$\frac{\nu}{\nu + X^2} \...