# 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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### Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
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### Help me understand the quantile (inverse CDF) function

I am reading about the quantile function, but it is not clear to me. Could you provide a more intuitive explanation than the one provided below? Since the cdf $F$ is a monotonically increasing ...
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### Why is the CDF of a sample uniformly distributed

I read here that given a sample $X_1,X_2,...,X_n$ from a continuous distribution with cdf $F_X$, the sample corresponding to $U_i = F_X(X_i)$ follows a standard uniform distribution. I have ...
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### Distribution of a ratio of uniforms: What is wrong?

Suppose that $X$ and $Y$ are two i.i.d. uniform random variables on the interval $[0,1]$ Let $Z=X/Y$, I am finding the cdf of $Z$, i.e. $\Pr(Z\leq z)$. Now, I came up with two ways of doing this. ...
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### Why the CDF for the Normal Distribution can not be expressed as a closed form function?

I am working my way through Think Stats, where the author states that "there is no closed form expression for the normal cumulative density function" but does not provide any further details as ...
586 views

### 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 ...
28k views

### How are the Error Function and Standard Normal distribution function related?

If the Standard Normal PDF is $$f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$$ and the CDF is $$F(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-x^2/2}\mathrm{d}x\,,$$ how does this turn into an error ...
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### Inverse function for a non-decreasing CDF

For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0<u<1.$$ Where U has a uniform $(0,1)$ ...
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### How to find quantiles and likelihoods of mixture distributions?

My PDF: M was estimated and found to be 5. I need to work out the quartiles for the PDF above. In addition, I need to use different methods of estimation to estimate the parameters. So far I've ...
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### Which to believe: Kolmogorov-Smirnov test or Q-Q plot?

I'm trying to determine if my dataset of continuous data follows a gamma distribution with parameters shape $=$ 1.7 and rate $=$ 0.000063. The problem is when I use R to create a Q-Q plot of my ...
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### How are percentiles distributed?

I was taking a look at this page, and I can't seem to understand why the frequency plot of the percentiles is uniformly distributed. Distances between percentiles are not equal, so why is the ...