# 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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### Simulating a joint distribution with the inverse method

I have the following joint distribution: $$f(x, y) = 3x^2y^xe^{-x^3}(1 + x),\quad x \gt 0,\ y \in (0,1).$$ I want to simulate a sample of this distribution through the inverse method but I don't know ...
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### How to compute the marginal CDF of a joint density?

I am trying to compute the cumulative distribution function of a random variable $u$ that has the following density: $$f(u) = \int_{1}^\infty \frac{e^{-4uv}}{v^5}dv$$ for $u \gt 0$. What I've tried ...
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### Deriving the CDF of student $t$-distribution

I am trying to derive the cumulative density function(cdf) of $t$-distribution, define as in https://en.wikipedia.org/wiki/Student%27s_t-distribution#Cumulative_distribution_function I derive it by ...
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### How to derive the distribution of 1 - X if X is Beta(0.5, 1)

I've been reading about uniform distributions and I'm wondering how statisticians derive these distributions. Lets say we have X, a random variable from the Beta(0.5, 1) density. How could you derive ...
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### How to prove $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$

I've been reading about uniform distributions but I can't see how $Y=X^{2}$ is Beta$\left(0.5,\ 1\right)$ if $X$ is Uniform$(0,\ 1)$. Is there a way to prove this using the cumulative distribution ...
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### stochastic dominant second and first order

I am a bit confused between first order stochastic dominant and second order stochastic dominant can you give me an example which it is second order dominant but not first order stochastic dominant
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### Generate random values using an empirical cumulative distribution function

I have a set of data points that I have used to generate my empirical CDF which looks like this (to simplify things I have reduced the number of points for this question but it shouldn't matter): ...
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### Generate random values using an empirical cumulative distribution function [duplicate]

I have a set of data points that I have used to generate my empirical CDF which looks like this (to simplify things I have reduced the number of points for this question but it shouldn't matter): ...
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### Is the CDF of the Mean always 0.5 for all kind of distributions?

Can we say that the value of the cumulative distribution function at the mean F(X< Mean) is always 0.5 for all kind of distributions (even ones that are not symmetric)?
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### How are these distributions same?

I am reading this paper 'Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations' and struggling with understanding the proof of their main claim (in Appendix A). ...
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### Finding Cumulative Distribution Functions and merging them

I made up a data set with n=314, mean =14.27854, standard deviation =2.16547 using p <-rnorm(314,14.27854, 2.16547). Now, I want to compare theoretical ...
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### How to derive the solution of $F_S(x)=P \left ({|h|^{2} \le \frac { x \left ({1 + |g|^{2} \rho _{2} }\right)}{\phi \rho _{1}} }\right)$?

[EDIT] I came across a received signal-to-interference-plus-noise-ratio (SINR), $S$, of a wireless communication system as \begin{align*} S = \frac{\phi|h|^2\rho_1}{1+|g|^{2} \rho _{2} }, \tag{1} \...
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### Question about the expression of the CDF for a standard normal distribution [duplicate]

My question is with regards to the second equation. Why is the term in the integral using the variable y? Shouldn't it be z? Because in the first equation, we have the PDF as a function of z. So to ...
1 vote
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### Question about the Solution for Problem 3.7b, Introduction to Probability (Bertsekas, 2nd Edition)

I am currently working on the problems in Introduction to Probability (Bertsekas and Tsitsiklis, 2nd Edition) and one of the problems is as follows: For Problem (b), the final answer is as follows: ...
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### Name of a distribution similar to the exponential

for a simulation I'm using the continuous distribution $$F(x)=1-(1+x)e^{-cx}$$ for $x\geq 0$ with $c\geq 1$. Do you know if this distribution has a name?
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### Understanding Simple Inverse Distribution

I know this is quite basic, but I fail to see where my mistake with the following simple example from wikipedia is. \begin{align*} G(y) &= \Pr(Y \leq y) \\ &= \Pr \left (X \geq \frac{1}{y} \...
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### Is there a simple method to produce a histogram of the Wilson distribution? [closed]

The Wilson score for some observed binomial distribution $(\hat{p},N)$ gives you a confidence interval on $p$. However, I need a histogram over the possible values of $p$. (With a relatively low ...
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### Sklar’s Extension Theorem and support restrictions

This question is about an application of the Sklar's Extension Theorem, whose proof can be found in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look ...
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### Examples of Defective Distributions

I was reading about Stochastic Convergence and I came across a term called Defective Distribution. Essentially what they refer to as a {Defective Distribution is a distribution that has all the ...
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### Kolmogorov Smirnov Test : CDF vs PDF

Does anyone know that why in the Kolmogorov-Smirnov Test, the empirical distribution function is compared with the cumulative distribution function and not the probability distribution function? Is ...