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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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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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Making Naive estimator/ cumulative function [closed]

I want to contruct a naive estimator X<-rnorm(100) and i’m interested in the function f(x)=1/2nh *sigma 1[x[i] is in between x-h and x+h] Where 1[.] above stands for indicator function and x[i]...
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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} \...
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How to approximate the student-t CDF at a point without the hypergeometric function?

Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF ...
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CDF Variable Transformation

Let $X$ be uniform on $(-1, 2)$ and let $Y = X^2$. Find the pdf of $Y$. So far I have noted that $F_X(x) = P(X \leq x) = \int_{-1}^x \frac{1}{3} dt = \frac{1}{3}(x+1)$. Then, since $Y=X^2$, $y \in [...
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When is the pmf of the difference of two independent random variables symmetric in zero?

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
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cumulative distribution function for non-normal distribution

From this article, I read that the author drew four versions of CDFs each plotted in different distributions (all four plots come from the same sample data) From these four plots, the author chooses ...
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Must the domain of a CDF be $\mathbb{R}$ or can it also be a strict subset?

So my question is whether the domain of a cumulative distribution function has to be $\mathbb{R}$ or whether it can also be a strict subset. The reason I'm asking is because I'm currently going ...
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Calculating a Confidence Interval for a Proportion for a Sample of Different Size

I'm interested in a (preferably analytic) solution or approximation to the following problem: Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
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How to see this order statistic result and find my error

Let W be a random variable with pdf $f(w)=\theta B^{-\theta}w^{\theta-1}$ for $0 \lt w \lt B$ and 0 otherwise. Assuming Independence, Show that , $W_{n:n} \to B$ as $n \to \infty$ where $W_{n:n}$...
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What does area under this density plot gives me?

I am new to data science and trying to grasp the concepts. I have a question in my exercise asking "what proportion of US states have populations larger than 10 million?". The density plot is shown ...
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Under what conditions does the two-sided DKW inequality become a strict equality?

If the two-sided DKW inequality is tight, then there should be a choice of distribution and sample size where the equality holds. What is it?
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What is the problem in my CDF derivation?

Let $Z = \frac{XY}{aX+bY+c}$ where the random variable $X$ and $Y$ follows gamma distribution such that $X\sim G(\lambda_x,\theta_x)$ and $Y\sim G(\lambda_y,\theta_y)$ The CDF of $Z$ can be ...
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CDF at n of normal distribution to the nth power

I'm working with an equation that includes a normally distributed investment return R. I can find the Cumulative Distribution Function of R for the first period n=0. However, how do I derive the CDF ...
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Probability of Ranking

Suppose that I have 3 normal distributions (I wish to extend this to K), such that $p_k\sim\mathcal{N}(\mu_k,\sigma_k^2)$. Assume that $(\mu_k,\sigma_k^2)$ are known. How would you go about ...
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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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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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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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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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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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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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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
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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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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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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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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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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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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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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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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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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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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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118 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
54 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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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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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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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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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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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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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 $X2$...