1. Introduction In the last post, we defined the convolution of the PDFs $f_{X}$ and $f_{Y}$ of two independent, continuous random variables $X$ and $Y$, and showed that it computed the ...

1. Introduction Suppose that $X$ and $Y$ are two continuous random variables. Given their PDFs PDFs $f_{X}$ and $f_{Y}$, how do you compute the $f_{X + Y}$ of the sum of $X+Y$? In th...

1. Introduction In the last post, we proved the following general formulae, after conducting some numerical experiments to gain intuition: \begin{align} F_{X_{(k)}}(x) &= \sum_{j=k}^{n} \bi...

1. Introduction In the last post, we defined the order statistics of a collection of iid random variables $X_{1}, \ldots, X_{n}$, to try and answer questions like “What’s the expected value of...

1. Introduction On average, how tall is the tallest 8th grader in a class of 30 students, in the US? What is the value of the average winning bid in a Vickery Auction, where the highest bidder w...

1. Introduction In the previous post, I recalled the definition of the cross entropy $H(p, q)$ of two discrete PDFs $p$ and $q$ over the same support $S = \{ x_{1}, \ldots, x_{n} \}$. It...

1. Introduction In this post, I discuss the cross entropy $H(p, q)$ of two discrete PDFs $p$ and $q$. It is a loose measure of similarity of $p$ and $q$, and so is used in machine lear...