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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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20 views

Normal Quantile Function With a lower bound not equal to infinity

I was recently at a statistics competition and a question came up as follows: They drew a normal distribution with $\mu=7$ and the area between the values $7.75$ and $8.25$ equal to $0.12$. No other ...
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17 views

Explanation for Cumulative Distributive Function example

I'd like to ask for clarification of the following example in my textbook. Example: Suppose events are occurring at random with average rate $\lambda$ per unit of time. What is the probability ...
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38 views

How do I fit a cumulative Gaussian distribution in R? [closed]

I am trying to fit a cumulative Gaussian distribution function to my data, but I'm not sure how to do this. From what I understand, the fitting process tries to find the mean and standard deviation of ...
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1answer
33 views

The distribution of a posterior predictive p-value under certain assumptions

I am wondering if anyone can check my understanding of the following passage concerning posterior predictive p-values in the textbook "Bayesian Data Analysis 3rd Edition" on page 151: In the ...
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1answer
33 views

Multi-dimensional CDF on a discrete support

Suppose I have two discrete-support random variables, $X$ and $Y$. They have joint CDF $F(X,Y)$. If I want to find $\Pr(a \leq X \leq b , c \leq Y \leq d)$. It is obviously not: $F(b ,d)-F(a-1 ,...
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38 views

PDF / CDF of sum of the product of constants with Nakagami-m random variables

I am trying to find the pdf (probability density function) and cdf (cummulative density function) of two Nakagami-m random variables multiplied by constants, e.g. ${(a_1 X_1)}^2$ + ${(a_2 X_2)}^2$. ...
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26 views

What is the probability that the new commitment will be fulfilled?

A consulting firm was hired to develop an Engineering project. Based on their previous experience, the direction of this office knows that the time (in months) needed to perform this type of task ...
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1answer
54 views

Computing the probability density function

Suppose we have the cdf $$F_X(x) = \begin{cases} 0 \quad \quad, x<-1 \\ 0.25 \quad \quad, -1\leq x < 1 \\ 0.5 \quad \quad, 1 \leq x < 2 \\ \frac{2}{3} \quad \quad, 2 \leq x < 3 \\ 1 \quad ...
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1answer
40 views

Evaluating problematic function when cdf is close to one?

Let $F(x;\theta)$ be a cumulative distribution function and $\beta>0$. I need to evaluate $$\rho=\frac{F(x;\theta)^\beta}{F(x;\theta)-F(x;\theta)^{\beta+1}},$$ but, for some values of $\theta$, R ...
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18 views

How to use multidimensional copula to obtain a joint distribution in python?

I am following this blog on how to use copula using python and scipy. From what I can understand, the process is as follows Generate samples from a multivariate distribution with a correlation (in ...
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2answers
40 views

How does this code find the CDF?

How does the below code give CDF? Can someone please explain what np.arange(len(sorted_data))/float(len(sorted_data)-1) does? ...
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25 views

Mean and variance of maximum of normal random variables

I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. Note that the $X_i$ are independent, but not identically distributed. That is, ...
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94 views

Distribution of maximum of normally distributed random variables

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$. My thought process so far: $$ \begin{align*} F_Y(y) &= \mathbb{P}(\max(...
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52 views

Cumulative distribution function of a squared laplace random variable

I am trying to calculate $F_Y(x)$ (CDF) of $Y=X^2$ where $X$ is a random variable of Laplace Distribution $f_X(x) = \frac{1}{2}e^{-|x|}$ (let's take a simple case when parameters $\mu=0$ and $b=1$). ...
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17 views

CDF of multiple experiment runs

I have an experiment in which I run multiple times with different seeds (10 in this case). As a result, I ended up with 10 different results. I know that if I want to calculate the total mean, I ...
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9 views

How this Equation is solved? How dBi is changed into rdr?

$Y_i = \frac{|h_{B_i}|^2}{1+d_{B_i}^\alpha}$ $d=distance, h_Bi=gain$ $f_{W_{B_i}}(\omega_{B_i}) = \frac{\lambda_{\Phi_B}}{\mu_{R_{D_B}}}=\frac{1}{\pi R_{D_B}^2} $ \begin{align} (CDF) of Y_i .... ...
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1answer
56 views

Quantile Function

I have seen the definition of quantile function here, which is as follows (slightly modified): Let $X$ be a real-valued non-degenerate random variable with distribution function $F_X(x)=\mathbb{P}({X\...
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1answer
27 views

How to generate a Weibull distribution with inverse transform

Given $X\sim \text{Weibull}(\lambda,k)$, generate samples from the Weibull distribution using the inverse transform. We know $F_X(x) = 1-\text{e}^{-(x/\lambda)^k}$ for $x\ge 0$ with $\lambda,k > 0$...
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1answer
71 views

CDF and PDF of radius of a unit disk

Let X and Y be uniformly distributed on a unit disk such that $x^2 + y^2 \leq 1$ Let $R = \sqrt{X^2 + Y^2}$. What are the CDF and PDF of $R$? I know that the area of the unit disk is $A = \pi r^...
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1answer
53 views

CDF and random variable

Please. I am trying to understand the proof, that cdf of minimum of $n$ random variables is $1-[1-F(x)]^n$ If I have $n$ independent random variables $X_1, \dots, X_n$, all of them have the same CDF $...
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58 views

Proving an inequality for CDF's

I am working on a proof to show that given $x_1, x_2,\ldots,x_k$ random variables with a joint pdf and joint CDF, show that $$ 1-\sum_{i=1}^k \overline{F_i(x_i)} \leq F(x_1,x_2,\ldots,x_k) \leq \min_i ...
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1answer
37 views

What is the distribution of $\frac{(Y_1 - Y_2)^2}{2},$ where $Y_i$ are standard Normal and independent.

Determine the distribution of $\frac{(Y_1 - Y_2)^2}{2},$ where $Y_i$ ~ $N(0,1),$ and $Y_1,Y_2$ are independent. I modelled the random variable in R and to me it seems like it's probably from a Gamma ...
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2answers
38 views

How to convert the parameters in a binomial distribution to those in a beta distribution?

I know that the beta distribution is the generalized continuous case of the discrete binomial distribution. Let's say I have a binomial distribution, $B(N,p)$. I would like to know the corresponding $\...
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need to transform a dataset but don't know how

Not a statistician by trade so my hands are tied. I have data from 6 populations (a1, a2, a3, b1, b2 b3) and have plotted the cumulative distribution plots (CDP) by one of the common features (say ...
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1answer
80 views

CDF of Piecewise Folded Normal

I came across a problem in a Carmona's Statistical Analysis of Financial Data in R (pg. 189, Problem 3.13). The due date has passed, so now it is considered a self-study question. I am seeking a ...
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24 views

How to calculate the average number of events per interval in poisson distribution

Hi dear statisticians, I have a random variable X that follows a poisson distribution. However, I only know the number of occurrences in an interval and the cumulative probability a. How I can ...
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31 views

Yn = min(X1 … Xn), X1 … Xn ~iid with pdf e^-(x-θ), X > θ . Why is the cdf of Yn = 1-e^-n(x-θ)

Yn = min(X1 ... Xn) X1 ... Xn ~iid with pdf e^-n(x-θ), X > θ The answer to a problem I couldn't figure out states that the cdf of Yn = 1-e^-n(x-θ) However, When I try to determine the cdf of Yn, I ...
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Evaluating the hazard function when the CDF is close to 1?

I need to evaluate a hazard function $h(t;\theta) = \dfrac{f(t;\theta)}{1-F(t;\theta)}$, where $f$ and $F$ are a pdf and a cdf, respectively, at many values of $t$ (and for several values of the ...
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Partial integration involving CDF [closed]

I am reading a textbook which claims that we can obtain by partial integration, for CDF $F(x)$:$$\int_{t}^{\infty} (1-F(x)) \frac{dx}{x}=\int_{t}^{\infty} (\log u -\log t) dF(u) $$ I am aware that ...
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Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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How to integrate probability density of sum of two indepedent random variables with a finite lower bound on one of them?

$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-u^2/2}\:du=1$$ but $$u = \ln(A)-C-k$$ where $\ln(A)$ and $C$ are normally distributed independent random variables, and $k$ is a constant. I am ...
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Finding probability of a point using bivariate copula density

I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
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217 views

How to find distribution function of sum of 2 random variables that are uniformly distributed? [duplicate]

I am stuck with this tutorial question in one of my stats module and I would greatly appreciate some help: Let $X1$ and $X2$ be independent random variables with $a = 0$ and $b = 1$ i.e. $X1$ and $X2$...
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2answers
119 views

Is this true: $P(X>x) = P(\log(X)>\log(x))$

I can't find any theorem regarding this. I know it works for normal / lognormal distributions and as the logarithm is an affine transformation and the cdf is increasing it seems plausible but i can't ...
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1answer
26 views

How can I find this constant?

My friend asked this question in our class: let X be a random variable which has a cumulative distribution function . Find (a). I think (a) cannot be solved but my other friend thinks (a) = 5/8 ...
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How do I get the CDF given a PDF with an indicator function?

Given $$f(x)=\frac{3}{4}(1-x^2 )𝟙_{(-1,1)}(x),\,x\in\mathbb{R}$$ How do I get $F(x)$? I know that the CDF of a random variable is defined as $$F(a)=\int_{- \infty}^{a} f(x)\,dx$$ In my case this ...
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1answer
193 views

Looking for an unbiased version of the empirical cumulative distribution function that I can interpolate

Most definitions of the ECDF define it as (#elements <= threshold) / #elements. Matlab and R both implement their ecdf() functions using this formula. In my testing, however, I find that there is ...
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1answer
48 views

Proof the joint bivariate cumulative distribution function

I would like to proof the expression ($P[X>x, Y>y]$) for two continuous random variables $X$ and $Y$. $P[X>x, Y>y]$ = $1- P[X \leq x, Y \leq y]$ (From the definition of probability). (...
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1answer
33 views

Why does the PDF use a different variable than x?

In the below image (from Wikipedia but also found in my text book), I noticed that the variable within the integrand is a "u" rather than the "x" which is found in the CDF function. Why is the ...
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Interpreting sklearn's GP R^2

Sklearn can compute an $R^2$ value of sorts for a Gaussian Process regressor. As explained here, the definition used is $1-u/v$, with $u$ ($v$) the residual (total) sum of squares. Since the result ...
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1answer
156 views

When we take draws from a normal distribution what are we drawing? [closed]

As I dig deeper than surface level in probability I'm starting to ask more questions I never thought about before. There are a bunch of intertwined concepts that are quickly becoming confused in my ...
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0answers
60 views

Computing the CDF of the minimum of particular dependent random variables

For each $i=1,\dots,n$ let $Z_i\sim\text{Poisson}(\lambda_i)$, and suppose $\{Z_i\}$ are independent. Also for each $i=1,\dots,n$, let $\{Y_{ij}\}_{j\in\mathbb{N}}$ be an infinite sequence of iid ...
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1answer
176 views

Logarithmic binning and log-normal distribution

I've an Italian cities dataset. It's similar to those British ones used in literature, but has some differences, though. I decided to perform a logarithmic binning to avoid noise on the right end of ...
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1answer
58 views

Joint CDF of dependent random variables: is knowing covariance sufficient?

Let $X,Y$ be real-valued random variables, which are dependent. Want: Calculate $\mathbb{P}[\,\min\{X,Y\}\leqslant0\,]$ (without Monte Carlo) Know: I can compute (numerically) $F_X$ and $F_Y$ (the ...
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21 views

Is the CDF of a time series fundamentally different (from a cross-sectional CDF)?

Shao (2015) has a strictly stationary univariate time series $X_t$ and he is interested in estimating $F_m$, the CDF of the $m$-period interval $Y_t=(X_t,...,X_{t+m-1})´$. To estimate the CDF, he has ...
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31 views

Joint distribution of multivariate normal

Let $X$ and $Y$ be i.i.d. $N(0, 1)$, and let $S$ be a random sign (1 or -1, with equal probabilities) independent of $(X, Y)$. \begin{align*} P((SX,SY)∈B)&=P((X,Y)∈B,S=1)+P((−X,−Y)∈B,S=−1) \\ &...
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2answers
47 views

Calculating multivariate integrals between lower and upper bounds

Suppose $\vec{X}=(x_1,x_2,...,x_n)$ follows some continuous multivariate distribution, such that $x_i\in{\rm I\!R}, i=1,...,n$. Suppose also that I have access to the following functions: $\phi(\...
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0answers
33 views

How to add dependence between random vectors using a copula?

I understand that copulas can be used as a tool to add any conceivable dependence to a pair of random variables. However, I would like to add some dependence between two random vectors. Let us ...
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1answer
296 views

In trouble with CDF graph

I'm trying to understand the meaning of this graph, which is a CDF (Cumulative Distribution Function). But I can't. Why is it starting from the top-left corner? I've never found such a graph. And ...
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1answer
68 views

finding quantiles of a kernel density estimation

I used R to find kernel density estimates of my dataset (for experiment I used 1000 samples generated from a known distribution in this step). I used code density()...