Probabilistic World

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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Probability: What Is It, Really?

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A ruler, a pen and a calculator on a notebook.Throughout history, we have come up with better and more accurate ways to measure physical quantities like time, length, mass, and temperature. This has been crucial for our scientific and technological development.

Each of these quantities has a precise definition and is informative about some aspect of the current state of the physical world. For example, the mass of an object can tell you how much work is necessary to lift it at a certain height. The outside air temperature determines the kind of clothes you would wear when you go out. And so on.

Probabilities are also quantities that measure something — they have a very precise and unambiguous mathematical definition. But still, they don’t relate to things in the physical world as straightforwardly and as intuitively as measures like mass and length.

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What Is a Sample Space?

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A photograph of two white and three brown large standard dice.The concept of a sample space is fundamental to probability theory. It is the set of all possibilities (or possible outcomes) of some uncertain process.

For example, the sample space of the process of flipping a coin is a set with 2 elements. Each represents one of the two possible outcomes: “heads” and “tails”. The sample space of rolling a die is a set with 6 elements and each represents one of the six sides of the die. And so on. Other terms you may come across are event space and possibility space.

Before getting to the details of sample spaces, I first want to properly define the concept of probabilities. [Read more…]