Beta distribution: trivial question

In summary, there is a way to go from the mean and variance of a beta distribution to the parameters α and β. This can be achieved by converting the formula for the mean into one for the ratio of β to α, and then simplifying the variance formula into two linear relationships between the parameters. However, this may result in at most three possible solutions, which can be eliminated based on the requirement that α and β must both be greater than 0.
  • #1
markuz
5
0
We all know that it is easy to get the beta mean and variance given the parameters α and β of the distribution (http://en.wikipedia.org/wiki/Beta_distribution).

Can we do right the opposite? I.e. is there any way to go uniquely from mean and variance to the parameters of the beta?
 
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  • #2
Based on the formulas in the link, the question can be reduced to solving a cubic equation in α or β as a function of the mean and variance. α and β are linear with respect to each other. Therefore there are at most three possible solutions. Since α and β are both required to be > 0, this could eliminate the extra solutions.
 
  • #3
Yes, there's a unique way. Convert the formula for mean into one for [itex]\frac{\beta}{\alpha}[/itex], then use that formula to simplify the variance formula into something linear in [itex]\alpha[/itex] and [itex]\beta[/itex]. Then you'll have two straight-line relationships (one positively sloped, one negatively sloped) between the two parameters.
 

FAQ: Beta distribution: trivial question

What is a Beta distribution?

A Beta distribution is a type of probability distribution that is often used in statistics to model random variables that take on values between 0 and 1. It is characterized by two parameters, alpha (α) and beta (β), which determine the shape and location of the distribution.

What is the purpose of using a Beta distribution?

Beta distributions are commonly used in statistical analysis because they can model a wide range of data, including proportions, probabilities, and percentages. They are also flexible and can take on a variety of shapes, making them useful in a variety of applications.

How is a Beta distribution different from other distributions?

Unlike other distributions, such as the normal or binomial distribution, the Beta distribution can take on a wide range of shapes, depending on the values of its parameters. It is also defined on a specific interval, typically between 0 and 1, whereas other distributions may be defined on an unbounded range.

How is a Beta distribution calculated?

The Beta distribution is calculated using the following formula: f(x|α,β) = (Γ(α+β)/Γ(α)Γ(β)) * x^(α-1) * (1-x)^(β-1). This formula involves the Gamma function, which is a special function used in mathematics to solve various types of problems involving continuous variables.

What are some real-world applications of the Beta distribution?

The Beta distribution has many practical applications, including in business, economics, and science. It is commonly used to model the success rate of a product or project, as well as to analyze the effectiveness of marketing strategies. It is also used in Bayesian analysis, which is a statistical technique used to update beliefs about a particular event or phenomenon as new data becomes available.

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