How Do You Calculate Probability?
My favorite part of mathematics is probability.
Last semester, I took a class in probability and random variables at MIT, and I got very interested in this field. The first time that I learned about probability-related problems was in lower school, when we learned about permutations and combinations, and a classic example could be the “poker hands,” such as calculating the probability of getting a “full house” and so on.
From my perspective, the most challenging part is to identify the overcounts during calculations. When taking this class, the emphasis was on random variables. There are two types of random variables: discrete and continuous. For discrete random variables, there are mainly four types: binomial, geometric, negative binomial, and poisson.
I was especially fascinated by the poisson random variables as well as the poisson point process.
For poisson random variables, the central idea is that if we know the frequency (or probability) of a certain event occurring during a set time period, then we could calculate the probability of any number of events happening in that time period with the poisson formula. Because these numbers are usually natural numbers, they are discrete random variables. On the other hand, continuous random variables also have four types: exponential, gamma, beta, and cauchy. I am most interested in exponential random variables, which has a classic example of the “bus problem”, that is, if we know the average waiting time for a bus to come, how long in expectation we would need to wait for the next bus to come.
When dealing with these types of problems, because they are continuous, normally we would need to calculate their pdf (probability density function) and cdf (cumulative distribution function).
Sometimes, if joint distribution happens, we will need to use the tool of multivariable calculus; in other cases, matrix manipulations could also come into role, and linear algebra can be implemented as well, and this is what I found most fascinating about mathematics: everything is interconnected, and learning the fundamental tools is very crucial in solving all kinds of problems in both mathematics and computer science.
Also, probability could have many other applications in machine learning. For example, the learning algorithms will make predictions and decisions based on probability. Probability is also closely related to statistics and data science. So it has countless potentials and applications, therefore, I find it very exciting and useful to learn.