Unlock the conditional expectation
An motivating example Suppose you have a wooden rod of a random length $L$, say uniformly distributed on $(0,1)$ (in meter). Then you will cut this rod a some point $Xin(0,L)$. Our question is what is the average of $X$? Given $L=ell$ , you would say that $mathbf{E}(X)=frac{ell}{2}$, isn’t it? It is true to think […]
You tested positive, but are you really sick? When a positive isn’t so positive: Uncovering the Math doctors often Overlook…
Introduction You tested positive. Your heart skips. The doctor’s tone shifts. But what does that result actually mean? Most people assume a positive test confirms the worst; that the disease is certainly present. In reality, the situation is far more subtle. Beneath the clinical certainty lies a quiet storm of probabilities, assumptions, and statistics. This […]
The strong independent segment lemma …
In this post, we’re going to talk about a minor result in complex analysis, but it was a clue for a main result in this paper. Riemann mapping theorem Riemann mapping theorem states that any two simply connected domains can communicate via a univalent function $f$. In other words, if $U$ and $V$ are […]
Taming the Beast: Conquer Differential Calculus with Confidence (Part two)
In our first discussion on differential calculus, we introduced the notion of differentiability for a multivariate function ( f ). In this note, we explore the concept in greater depth. Let $$ f(x_1, dots, x_n) = f(mathbf{x}) $$ be a real-valued function defined on some open subset ( U subset mathbb{R}^n . Definition. The partial […]
Taming the Beast: Conquer Differential Calculus with Confidence (Part one)
Differential Calculus, what’s it About? Differential calculus can be thought of as the study of how a small change (or perturbation) in the input affects the output. For instance, suppose a factory is set up to manufacture coins with radius (1) cm, but due to a machine’s imprecision, it actually produces coins of radius (1 […]
Randomness and probability …
On randomness According to Oxford Dictionary, the word “Randomness” refers to the apparent or actual lack of definite pattern or predictability in information.pause That is, most phenomena in life are random. For example : • Weather condition. • Gold price. • Reaction of your wife after giving her a gift. • etc … Definition 1. […]
