# Tag Info

Accepted

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

Convolution calculations associated with distributions of random variables are all mathematical manifestations of the Law of Total Probability. In the language of my post at What is meant by a “...
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### Intuitive explanation of Kolmogorov Smirnov Test

The Kolmogorov-Smirnov test assesses the hypothesis that a random sample (of numerical data) came from a continuous distribution that was completely specified without referring to the data. Here is ...
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### Why is the sum of two random variables a convolution?

Notation, upper and lower case https://en.wikipedia.org/wiki/Notation_in_probability_and_statistics Random variables are usually written in upper case roman letters: $X$, $Y$, etc. Particular ...
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### Why is the sum of two random variables a convolution?

Your confusion seems to arise from conflating random variables with their distributions. To "unlearn" this confusion, it might help to take a couple of steps back, empty your mind for a moment, ...
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### Empirical CDF vs CDF

Let $X$ be a random variable. The cumulative distribution function $F(x)$ gives the $P(X \leq x)$. An empirical cumulative distribution function function $G(x)$ gives $P(X \leq x)$ based on the ...
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### Distribution function terminology (PDF, CDF, PMF, etc.)

As noted by Wikipedia, probability distribution function is ambiguous term: A probability distribution function is some function that may be used to define a particular probability distribution. ...
• 140k
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### How to calculate the probability of a data point belonging to a multivariate normal distribution?

Yeah, that sounds right. If you have parameters $\mu$ and $\Sigma$ and data point $x$, then the set of all data points that are less likely than $x$ are the ones that have less density, or in other ...
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### Why do we use parametric distributions instead of empirical distributions?

An enormous amount of data is needed to accurately estimate a distribution nonparametrically, especially a continuous one. Even then, some assumptions about the smoothness of the distribution are ...
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### How can this CDF be decreasing? (power-law)

The vignette by Gillespie (2020) cites Clauset et al (2009) as another source of the figure. Reading that latter paper, it looks like they are actually plotting the complementary CDF (better known as ...
• 129k
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### On the proof of right-continuity of the distribution function

First of all, let's write the desired result mathematically: $F$ is right continuous-means really $F(x) = F(x^+)$ for all $x$, where $$F(x^{+}) = \lim_{y\to x^+} F(y).$$ I'm going to assume you are ...
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### Why is the CDF of a sample uniformly distributed

Here's some intuition. Let's use a discrete example. Say after an exam the students' scores are $X = [10, 50, 60, 90]$. But you want the scores to be more even or uniform. $h(X) = [25, 50, 75, 100]$ ...
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### Empirical CDF vs CDF

The empirical CDF is built from an actual data set (in the plot below, I used 100 samples from a standard normal distribution). The CDF is a theoretical construct - it is what you would see if you ...
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### Why is cumulative distribution function monotone non-decreasing?

Because if $x \leq y$, then if $X \leq x$, it follows that $X \leq y$. Therefore, $P(X \leq x) \leq P(X \leq y)$. More generally, probabilities are monotone in the sense that if $A$ and $B$ are events ...
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• 81.5k
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### Examples of distributions with easily solvable quantile functions but hard to solve CDFs

For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$,...

### Why is the empirical cumulative distribution of 1:1000 a straight line?

The cumulative distribution function of a random variable $X$ has nothing to do with summing the random variable. It is the probability that $X$ will take a value less than or equal to $x$. And of ...
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Let $x$ by any number. Consider the event $\min(X,Y)\le x$. It can be expressed as the union of two events $$\min(X,Y)\le x = (X\le x) \cup (Y \le x),$$ shown by the overlapping yellow and green ...