Probabilistic World

Cryptography: Historical Introduction & Combinatoric Analysis (Series)

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Cryptography, historical series, part 1 (the Kryptos sculpture in the background)

Hi everyone, and welcome to my series on cryptography!

Initially, I started writing this as a single post, but as I was planning what I wanted to include, and seeing how long it would become, I decided it’s best to split it into separate posts.

Cryptography is the study of techniques and procedures, such as using codes and ciphers, for secure (secret) communication between two or more agents. More specifically, it deals with finding sophisticated ways of hiding the true content of messages exchanged between communicators. People use cryptography in military and diplomatic communication, password protection, securing online transactions, as well as in any other context where some secret information needs to be protected from being accessed or understood.

As you’ll see in future posts, probability theory too can play an important role in cryptography.

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Discrete Probability Distributions: Overview (Series)

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A discrete probability distribution, two hands holding dice, and a background referencing the movie The Matrix

In my previous two posts I sketched the frame of the big picture around probability distributions. In my introductory post I gave some intuition about the general concept and talked about the two major kinds: discrete and continuous distributions. And in the follow-up post I related the concepts of mean and variance to probability distributions. I showed that this connection itself goes through two fundamental concepts from probability theory: the law of large numbers and expected value.

Now I want to build on all these posts. My plan is to start introducing commonly used discrete and continuous distributions in separate posts dedicated to each one. And I want to start with the former, since they are significantly easier to understand.

The goal of the current post is to be a final warm-up before delving into the details of specific distributions.

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Mean and Variance of Probability Distributions

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In my previous post I introduced you to probability distributions.

In short, a probability distribution is simply taking the whole probability mass of a random variable and distributing it across its possible outcomes. Since every random variable has a total probability mass equal to 1, this just means splitting the number 1 into parts and assigning each part to some element of the variable’s sample space (informally speaking).

In this post I want to dig a little deeper into probability distributions and explore some of their properties. Namely, I want to talk about the measures of central tendency (the mean) and dispersion (the variance) of a probability distribution.

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Introduction to Probability Distributions

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If you want to take your understanding of probabilities to the next level, it’s crucial to be familiar with the concept of a probability distribution.

In short, a probability distribution is an assignment of probabilities or probability densities to all possible outcomes of a random variable.

For example, take the random process of flipping a regular coin. The outcome of each flip is a random variable with a probability distribution:

  • P(“Heads”) = 0.5
  • P(“Tails”) = 0.5

Depending on the type of random variable you’re working with, there are two general types of probability distributions: discrete and continuous. In this post, I’m going to give an overview of both kinds. And in follow-up posts I’m going to individually introduce specific frequently used probability distributions from each kind.

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The Variance: Measuring Dispersion

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Small pawn-shaped magnets of different color, representing varianceA few posts ago I introduced you to the “three M’s” of statistics — the concepts of mean, mode, and median. Today I want to talk to you about a related concept called variance.

While the three M’s measure the central tendency of a collection of numbers, the variance measures their dispersion. That is, it measures how different the numbers are from each other.

Measuring dispersion is another fundamental topic in statistics and probability theory. On the one hand, it tells you how much you can trust the central tendency measures as good representatives of the collection. High variance usually means a lot of the numbers in the collection will be far away from those measures.

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Intuitive Explanation of Expected Value

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A pile of poker chips and a few dice

Expected value is another central concept in probability theory. It is a measure of the “long-term average” of a random variable (random process). I know this doesn’t sound too clear, but in this post I’m going to explain exactly what it means.

There are many areas in which expected value is applied and it’s difficult to give a comprehensive list. It is used in a variety of calculations by natural scientists, data scientists, statisticians, investors, economists, financial institutions, and professional gamblers, to name just a few.

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