Beta distribution

As explained in Grokking Statistics

A continuous probability distribution over probabilities of success, parameterized by the observed numbers of successes and failures.

A beta distribution describes which underlying success rates are most likely after observing successes and failures. With 7 successes and 3 failures, alpha = 7 and beta = 3. section 3.8

Unlike the binomial distribution, which models a count of successes, the beta distribution models the continuous range of possible success rates from 0 to 1. Its smooth curve shows probability density, not probability directly; the total area under the curve is 1. section 3.12 section 3.8

Estimating a success rate

Suppose a manufacturer records 7 successes and 3 failures. Use SciPy to draw the beta density and calculate the area between success rates of 0.4 and 0.7: from scipy.stats import beta a, b = 7, 3 area = beta.cdf(0.7, a, b) - beta.cdf(0.4, a, b) print(area) # 0.437796414 The result is the area under the curve between 0.4 and 0.7, providing a way to examine how plausible that range of underlying success rates is given the observations. section 3.8

Figure 3.6 The beta distribution showing probabilities of success rates for a single trial

View the smooth beta-distribution curve for the possible success rates. section 3.8

How Grokking Statistics covers it

How deep it goes

Explains parameters, continuous density curves, unit area, and interval areas calculated with a cumulative distribution function.

Examples

Uses 7 successes and 3 failures, a SciPy density plot, and the interval from 0.4 to 0.7.

What it assumes

Assumes basic success-and-failure trials while teaching alpha and beta as observed successes and failures.

What you can do after

Apply it via SciPy to plot success-rate density and calculate the area within a chosen probability range.

How it connects

Connects success-rate estimation with binomial distributions, Bernoulli trials, probability density functions, and Monte Carlo simulations.

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This explainer was generated by Manning's knowledge graph from the text of Grokking Statistics. The section links point to the parts of the book it draws on.