Beta Distribution - Examples, Formula and Applications (2024)

FAQs

What is an example of a beta distribution? ›

The beta distribution is a continuous probability distribution that can be used to represent proportion or probability outcomes. For example, the beta distribution might be used to find how likely it is that your preferred candidate for mayor will receive 70% of the vote.

Also, the beta distribution is used in PERT where it produces a bell-shaped curve which is nearly normal. Consider an example; the beta distribution is used in the Bayesian Analysis to define the initial knowledge about the probability of the success that helps to complete the specified mission successfully.

Beta Derivation

The Probability Density Function (PDF) for X ∼ Beta ( a , b ) is: f ( X = x ) = { 1 B ( a , b ) x a − 1 ( 1 − x ) b − 1 if 0 < x < 1 0 otherwise where B ( a , b ) = ∫ 0 1 x a − 1 ( 1 − x ) b − 1 d x A Beta distribution has E [ X ] = a a + b and V a r ( X ) = a b ( a + b ) 2 ( a + b + 1 ) .

Examples of Beta

As an example, consider an electric utility company with a β of 0.45, which would have returned only 45% of what the market returned in a given period. Negative β – A company with a negative β is negatively correlated to the returns of the market.

The beta distribution is used to model continuous random variables whose range is between 0 and 1. For example, in Bayesian analyses, the beta distribution is often used as a prior distribution of the parameter p (which is bounded between 0 and 1) of the binomial distribution (see, e.g., Novick and Jackson, 1974).

Beta particles can also be used to treat certain types of cancer. In manufacturing, beta particles are used to test the thickness of paper. If the paper is thicker, less beta particles will be able to pass through it. In this way, beta particles can be used for quality control.

Beta Test Examples

Another example is when a hardware manufacturer sends pre-release units of a new gaming console to a select group of gamers for beta testing. These gamers play various games on the console, test their performance, and provide feedback on any issues they encounter.

Beta function helps describe the shapes of probability curves in statistics. It's a type of curve used to model things like proportions or probabilities. Beta function helps update beliefs with new data, especially in fields like finance or medicine.

Beta distributions are useful because they are defined between a minimum and maximum value, many different shapes are available depending on the parameters chosen, and, for some parameter choices, the distribution has a well defined mode (most likely point).

What is the industrial use of beta? ›

In paper manufacturing, beta gauges are used to monitor the thickness of the paper at speeds of up to 400 m/s.

Specifically, imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, then two black balls are returned to the urn.

Properties of Beta Distributions

If X∼beta(α,β), then: the mean of X is E[X]=αα+β, the variance of X is Var(X)=αβ(α+β)2(α+β+1).

How to calculate a stock's beta. A stock's beta is equal to the covariance of the stock's returns and its benchmark index's returns over a particular time period, divided by the variance of the index's returns over that period. As a formula, β = covariance(stock returns, index returns) / variance(index returns).

Beta can be calculated by dividing the asset's standard deviation of returns by the market's standard deviation. The result is then multiplied by the correlation of the security's return and the market's return.

What are some common sources of beta particles? Many beta emitters occur naturally in the radioisotopes found in the natural radioactive decay chains of uranium, thorium and actinium. Examples include lead-210, bismuth-214 and thallium-206.

Another example is when a hardware manufacturer sends pre-release units of a new gaming console to a select group of gamers for beta testing. These gamers play various games on the console, test their performance, and provide feedback on any issues they encounter.

For example, the beta diversity between the woodland and the hedgerow habitats is 7 (representing the 5 species found in the woodland but not the hedgerow, plus the 2 species found in the hedgerow but not the woodland). Thus, beta diversity allows us to compare diversity between ecosystems.