Tag: theorem

The mean value theorem for integrals relates the area under a curve (the definite integral) to the mean value of that curve over the same interval. It is quite a simple theorem, in fact almost obvious, but other important theorems rely on it. We will start with a graphical illustration before mov...

Out of the many techniques there are for solving limits, the squeeze theorem is a fairly famous theorem that has the ability to evaluate certain limits by comparing with other functions. For those who do not know the squeeze theorem, it states the following: Let A be some domain contain...

Now draw a line — its length equal to the diameter of the circle — through its center: The length of AB is equal to the diameter of the circle Choose a point somewhere on the circle (but not on the line), and call it C. And finally, draw lines AC and ...

Recently I started reading the wonderful book, “The Mathematical Mechanic” by Mark Levi, in which he has proved a wide range of mathematical theorems using purely physical arguments. Inspired by some of these theorems, I decided to prove Viviani’s theorem from a physics perspect...

Introduction: In the vast realm of probability and statistics, Bayes’ Theorem stands as a cornerstone, offering a profound understanding of conditional probability. Named after Reverend Thomas Bayes, an 18th-century statistician and theologian, this theorem provides a systematic framework f...

In the realm of data science, machine learning, and data analysis, having the ability to make decisions based on trustworthy evidence is vital. Bayes’ Theorem, a fundamental concept in probability theory, offers a robust framework for updating beliefs and making predictions based on new i...

Imagine you’re on a treasure hunt, but instead of a map, you have clues that change as you find more treasure. This is similar to Bayes’ Theorem, a mathematical formula that helps you update the probability of an event as you get new evidence. It’s like a guide in making decisions ...

The Central Limit Theorem (CLT) is fundamental in statistical fields and probability theory, and allows us to make quantitative probability statements when given sufficiently large enough number of samples, under the standard normal distribution. It is a beautiful distribution that fundamentally occ...

A probability experiment Imagine you have a bag containing 100 plastic tokens. Some are square, some are circular. Some are red, some are blue. To be precise: 80% of the tokens are squares, the rest are circles. Of the square tokens, 75% are red and 25% are blue. Of the circle tokens, 1...

Probability is a fundamental concept in the realm of statistics and decision-making. It allows us to quantify uncertainty and make informed judgments based on available information. Among the many tools in the probability theory, Bayes’ Theorem stands as a cornerstone. Baye’s theorem ...

Unraveling the Power of Bayes’ Theorem: A Window into Probabilistic Reasoning Introduction: In the vast realm of probability and statistics, Bayes’ Theorem stands as a cornerstone, offering a profound understanding of conditional probability. Named after Reverend Thomas Bayes, an 1...

Theory Let Xᵢ with i = {1 … n} be n random variables (r.v.), independently distributed with distributions Dᵢ not necessarily identical, but with the same mean μ and variance σ². (1) Let us define the “...

A couple of nights ago I was lying in bed watching some lectures, reading some articles, and so on, and at some point, I decided what I thought to be a wise decision at the time, namely that I would only watch one more math video before going to sleep. I had absolutely no clue that the ...