One fine summer morning not so long ago, I woke up with a start to the stark realization that I had no idea how to derive the formula for the sum of powers of natural numbers. I am talking about a closed-form expression for the series (for non-negative integer \(p\)) \[ \sum_{i=1}^n {i^p} = 1 + {2^p} + {3^p} + \dots + {n^p} \]
Last week, for pedagogical reasons, I had to solve the following problem:
In a bag there are \(4\) blue and \(5\) red balls. We randomly take \(3\) balls from the bag without replacement. Let the random variable \(X\) denote the number of blue balls we get. Find out the expectation and variance of \(X\).
Linear regression is possibly the most well-known machine learning algorithm. It tries to find a linear relationship between a given of set of input-output pairs. One notable aspect is that linear regression, unlike most of its peers, has a closed-form solution.
The mathematics involved in the derivation of this solution (also known as the Normal equation) is pretty basic.
I have been running this site on WordPress for the past one month. Just after the first week, I started getting doubts about the future of our relationship. It is not that WordPress lacks power or expressiveness. One could build almost any kind of site using WordPress with very little effort.
Years back, I encountered the following problem in a book on Number Theory: Prove that the sum \(H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\) is never an integer after \(n > 1\). In other words, in the sequence \(1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots\), there is only one integer, and that is the first number \(1\).
A famous problem in probability is this: Suppose you randomly1 break a stick in two places. You will get three smaller pieces of stick. What is the probability that you can form a triangle using these pieces?
One interesting thing about this problem is that depending on how you visualize breaking the stick, you get different probabilities.