{"id":3295,"date":"2022-01-05T08:16:00","date_gmt":"2022-01-05T07:16:00","guid":{"rendered":"https:\/\/www.gironi.it\/blog\/?p=3295"},"modified":"2026-03-01T20:01:14","modified_gmt":"2026-03-01T19:01:14","slug":"the-beta-distribution-explained-simply","status":"publish","type":"post","link":"https:\/\/www.gironi.it\/blog\/en\/the-beta-distribution-explained-simply\/","title":{"rendered":"The Beta Distribution Explained Simply"},"content":{"rendered":"\n

The Beta distribution is a crucial probability distribution in Bayesian statistics<\/a>.<\/p>\n\n\n

In theoretical probability problems, we know the exact probability value of a single event, making it relatively straightforward to apply basic probability calculation rules to reach the desired result.<\/p>\n\n\n

In real life, however, it’s much more common to deal with collections of observations, and it’s from this data that we must derive probability estimates<\/strong>.<\/p>\n\n\n\n\n\n

To put it more clearly: in life, we almost never have access to the exact probability value of an event: rather, we have data and observations.
Deriving probabilities from observed data is what we call statistical inference<\/strong>.<\/p>\n\n\n

Beta is a continuous value distribution<\/strong>, and in this respect, it differs from the binomial<\/strong><\/a> distribution, which, as we’ve seen, presents discrete values.<\/p>\n\n\n

We define it through a probability density function<\/strong> (PDF<\/strong>): (no, not the well-known format created by Adobe…<\/em>)<\/p>\n\n\n\\(\nBeta(p;\\alpha,\\beta)=\\frac{p^{\\alpha-1} \\times (1-p)^{\\beta-1}}{beta(\\alpha;\\beta)} \\\n\\)\n\n\n

where<\/p>\n\n\n

p<\/em> = is the probability of an event
\u03b1 = how many times we observe our event of interest
\u03b2 = how many times our event of interest does NOT occur
and obviously:
\u03b1 + \u03b2 = number of trials<\/p>\n\n\n

The beta function (not the \u03b2 value) in the denominator serves to normalize the result<\/strong> (which will thus be between 0 and 1).
It is calculated through numerical integration, since the distribution is continuous.<\/p>\n\n\n

The Beta distribution is a probability distribution of probabilities<\/strong>, and since it models a probability, its domain is limited between 0 and 1.<\/p>\n\n\n

\"Fantasy<\/figure><\/div>\n\n\n

Let’s look at a practical example of the beta distribution using R<\/h2>\n\n\n

Imagine that an online game organizer claims that at least 1 in 10 players wins a prize. We have the data, and we know that among the last 800 players, there were 65 winners.<\/p>\n\n\n

The question we ask ourselves is: is the game organizer telling the truth based on our data? Can we consider that a player has at least a 10% chance of winning a prize when buying a ticket based on our sample?<\/p>\n\n\n

The solution to our question can be easily derived using the beta function with our data:<\/p>\n\n\n

We use the cumulative beta distribution:
\u03b2 (.1, 65, 735, TRUE)<\/p>\n\n\n

In R, it takes just one line to find the part of our function that lies between 0.1 and 1, showing the probabilities above 10% of winning a prize when buying a ticket:<\/p>\n\n\n

integrate(function(x) dbeta(x,65,735),0.1,1)\n\n0.03170546 with absolute error < 2.3e-06<\/pre>\n\n\n
The answer is right before our eyes. The probability of having at least 10% success is just 3.17%. What the game organizer claims, in light of the data, is false.<\/p>\n\n\n
You might also like<\/h2>\n\n\n