# 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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### How to write down a cumulative distribution function that consists of two distributions

I am being asked to write a CDF of a random variable $X$. I know that there is $0.5$ probability that $X=5$ and $0.5$ probability that $X$ follows the exponential distribution with parameter $7$. I ...
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### How to Solve For the Inverse Cumulative Distribution Function of a Double-Exponential Probability Density Function

I'm stuck on figuring out how to sample data from a fake/known double-exponential PDF for a lab project involving C. elegans egg-laying rate data. I need help with figuring out if there's an exact ...
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### Meaning of uniform percentile rank distribution?

I'm new to statistics and I've been following Think Stats 2. I've just gotten to Cumulative Distribution Functions. I have some questions regarding uniform distribution: What does it mean for the CDF ...
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### How to properly implement Student's T CDF efficiently

X-post from SO: The longshot here is to use this function for the sake of calculating accurate P values in pinescript using the student's t distribution. My attempt to replicate this implementation is ...
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### About unique determination of symmetric point (or center) of a distribution based on pdf or cdf

Suppose we have a distribution that is known to be continuous and symmetric, and is otherwise unknown. We want to decide whether it is actually centered at zero using an equation involving pdf or cdf. ...
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### How to prove that a function is 2-increasing (copula)

There are three conditions to prove that a function is a copula: $C(u,0)=0=C(0,v)$ grounded. $C(u,1)= u, C(1,v)= v$. $C(u,v)$ 2-increasing function. Here I am concerning in the last condition how to ...
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### Is the definition of symmetric distribution using cdf correct?

Based on wikipedia (https://en.wikipedia.org/wiki/Symmetric_probability_distribution), a distribution is symmetric about $x_0$ if and only if it is a distribution whose pdf(or pmf) $f(\cdot)$ ...
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### point such that area to the right of that point under one gaussian is 5% of area under a second gaussian

Say I have two gaussian random variables $Z_1 \sim f_1 = f(\cdot|\mu_1, \sigma_1)$ and $Z_2 \sim f(\cdot|\mu_2, \sigma_2) = f_2$, where $f$ is the gaussian density. How can I calculate the value of $x$...
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### How to compute the median of a continuous distribution?

I don't have a solid background in statistics so the concept of probability density functions in the statistics course I'm taking is new to me. I need to derive the median of a continuous distribution ...
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### Creating a Probability Plot of a Custom Distribution

Let's say we have some icdf function, which I will paste below: ...
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### Which $\mu$ hold so that integral of CDF (from $\mu$ to $\infty$) equals to integral of 1-CDF (from $-\infty$ to $\mu$)?

What is the $\mu$ s.t. $$\int_{\mu}^{\infty}1-F(x)dx = \int_{-\infty}^{\mu}F(x)dx?$$ Here $F(x) = P(X\leq x).$ Should $\mu$ be the median of X, i.e. $0.5=F(\mu)$? I think $\mu$ should be the point so ...
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### Copula between a distribution and its univariate transformation

I'm trying to compute the copula (or joint distribution) between x and a univariate transformation, like say sin(x). That is compute $C_{XY}$ (or $F_{XY}$) given that $x \sim U(0,1)$ and $y = sin(x)$ ...
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### What is the ppf of the truncated normal distribution?

What is the percent point function (ppf), or inverse cdf, of the truncated normal distribution? The distribution and cdf is defined here: https://en.wikipedia.org/wiki/Truncated_normal_distribution ...