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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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 ...
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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).
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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, ...
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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 ...
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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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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:...
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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 ...
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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, ...
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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 ...
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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}}...
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Copula Invariance Principle
I don't get why equation 7 is true, can someone explain me why? This is part of the proof of the invariance principle in copula theory.
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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 ...
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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
$$X = f(A,C) = 2A + 3B + 4C = 2A + 3(...
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Proper Bivariate Joint Survival function
I have a joint survival function in the presence of 2 competing risks.
$S(t_1, t_2) = \exp\{−λ_1t_1 − λ_2t_2 − νt_1t_2 − μ_1t_1^2
− μ_2t_2^2\} \ \ \ t_1,t_2\geq0$
For which parameter values is this a ...
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Approximate distribution of random variable similar to studentized mean R.V?
It is well known that the distribution of the studentized mean, i.e., $T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} $, can be approximated (...
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Problem with deriving the cumulative distribution from the density function
Consider the continuous distribution with density function
$$
p(x) = \frac{1}{2}\cos(x) \;, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}.
$$
I want to derive the cumulative distribution function for ...
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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 ...
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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, ...
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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 ...
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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) ...
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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 + ...
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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 \...
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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 ...
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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.
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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 ...
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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 ...
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Parameters of the log-normal from CDF of a composition of $n$ i.i.d
Let $X_1,\ldots,X_n$ be i.i.d. log-normal random variables such that $$\log(X_i)\sim N(\mu,\sigma^2)\ \ \forall i=1,\ldots,n$$
Now let $Y$ be equal to the $\min(X_1,\ldots,X_n)$. It is quite easy to ...
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Exact confidence interval for the binomial distribution
I am confused about the way to obtain an exact confidence interval for the binomial distribution.
Say, we have $n$ samples, $X_1, X_2, ..., X_n \in \{0,1\}$. Then the probability of having $k$ ...
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Determine CDF and PDF from quantiles
I would like to determine CDF and PDF from quantiles that I have determined via quantile regression.
I have read here in the forum (Find the PDF from quantiles) that it is possible to interpolate this ...
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The probability integral transform when the CDF is non-decreasing
I'd like to ask about middle of the discussion in this answer. Writing things out in reverse, $\mathrm{Prob}(F_X(X) \leq y) = \mathrm{Prob}(X \leq \mathrm{inf}\{x: F_X(x) \geq y\})$. Why is $\mathrm{...
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Integral of cdf of a symmetric random variable
How to compute $$\int_{-k}^{k}F(x)dx$$
where $F(x)$ is the cumulative distribution function of continuous random variable $X$ which has symmetric pdf about $x=0$ and $k>0$.
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Statistical Inference by Casella Exercise 4.51 [duplicate]
I am self-studying statistical inference by Casella and Berger and having difficulty solving exercise 4.51:
let $X, Y, Z \sim U(0,1)$ and they are independent. Find $P(X/Y \leq t)$ and $P(XY \leq t)$. ...
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Cumulative distribution function of mixed variables
Given the probability density function:
$
\begin{equation}
f_{X, Y}(x, y)=\begin{cases}
\frac{xy}{3}, & \text{if } x=1,2,3 \text{ and } 0 < y < 1.\\\\
0, & \text{otherwise}.
\...
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How do we define the pdf in the multi-variate case and compute expectations?
Apologies if this is a very simple question but trying to work through a result in a paper made me realize I missed something a bit fundamental in my undergrad probability and analysis courses. Lets ...
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How to show that the negative binomial CDF converges to the Poisson CDF? (Incomplete beta vs incomplete gamma functions)
Question: Is there a straightforward proof of the following relationship between the (lower, non-regularized) incomplete beta function $\mathcal{B}(x; a ,b)$ and the (upper, non-regularized) ...
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Cumulative Distribution Function of the skewed generalized error distribution
I´m trying to calculate the cumulative distribution function of the skewed generalized error distribution with the probability density function:
where u = y - m.
From Theodossiou in (https://www....
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Find the mean of distribution given its cdf [duplicate]
P[X<y]=integral from 1 to y of (1/x)dx
how to find mean of this distribution?
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On the proof of right-continuity of the distribution function
In today's statistics class, we saw properties of the distribution function, i.e. defined by $F(x) = P(X\leq x)$ for a random variable $X$. One of these was:
$F(x)$ is right continuous.
The proof ...
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cmprsk cumulative incidence - comparing between two curve with different outomes
I have a question regarding the extracting p-values from the cumulative incidence curves that considered competing risks.
I used cmprsk packages and ...
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In statistics how does one find the mean of a function w.r.t the uniform probability measure?
I am unfamiliar in statistics. My knowledge is in pure mathematics.
Suppose $n\in\mathbb{N}$, where $X$ is in the $\sigma$-algebra of Caratheodory-measurable sets such that $X\subseteq\mathbb{R}^{n}$ ...
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What is the probability the expected value is undefined or infinite?
What is the probability from a uniform probability measure (pg.37) on sample space $\left\{N(\theta,1)|\theta\in[0,1]\right\}$ that for some random variable $X$ in the sample space, the Expected-Value ...
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How to combine two integrals containing the PDFs of a variable and its linear transform?
Original Post:
Suppose we have two random variables $X$ and $Y$ with cumulative distribution functions $F(x)$ and $G(y)$. We know that $Y = aX + b$.
I want to compute $Z(x) = F(x) - G(y)$.
What I have ...
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PDF does not integrate to 1 - where is my mistake?
I am trying to solve a question which gives me a random variable with the distribution function below
$$
F(x) = 1 - \left(\frac{\mu}{x}\right)^{2n}
$$
where $0 < \mu \le x < \infty$
I ...
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Help with a proof regarding empirical CDF
Given $X_1,X_2...X_n \sim F$ and $(\hat{F}_n(x))$ is an empirical CDF
I need to prove that $$Var(\hat{F}_n(x)) = \frac{F(x)(1 - F(x))}{n}$$
So what I did is:
The variance of an estimator $\hat{F}_n(x)$...
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Integral w.r.t. a cumulative density function
I was reading a paper and couldn't understand the following transition. Could someone tell me where the term of $p^k (\frac{1}{2} − c′)$ comes from in the following transition?
Def: Cumulative ...
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CDF Graph looks weird [closed]
I have this piecewise pdf function:
with the CDF:
When I graph the CDF above, I get this graph which looks weird for a CDF and I dunno how to explain what is happening. It looks more like an inverse ...
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Numerical Stability when Inverse CDF Sampling from Truncated Density
Let $f(x)$ be the pdf of a random variable that we want to truncate to the interval $[a,b]$ and then sample from it. Let $F(x)$ denote the corresponding cdf. We can use inverse cdf sampling and ...
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Asymmetries in the DKW bound
Suppose I have $n$ i.i.d. samples $X_1,...,X_n$ drawn from a distribution with CDF $F$. We use the samples to form the empirical CDF:
$$F_n(x)=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{X_i\leq x}$$
The ...
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Derivation of the LogNormal CDF from PDF [duplicate]
I've been trying to derive the CDF of the lognormal distribution.
I got this far but now I'm stuck.
$F(x) = \frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x\frac{1}{z}e^{-t^2}dz$
where $t = \frac{\ln(z)-\...
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Finding random number based on a specific discrete distribution using R
I have a discrete distribution with cumulative function:
$$
F(x)=Pr(X \leq x) = \frac{1}{(x-1)!}
\left( \Gamma(x,\lambda)-\frac{\lambda^{x-1} \exp(-\lambda)}{\lambda+1} \right)
$$
for $x=1,2,3,\ldots ....
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