[Maths Class Notes] on Beta Distribution Pdf for Exam

The Beta Distribution is the type of the probability distribution related to probabilities that typically models the ancestry of probabilities. The Beta curve distribution is a versatile and resourceful way of describing outcomes for the percentages or the proportions. As the Beta Distribution basically represents the probability, its domain is restricted between 0 and 1. For instance, what is the possibility of Vladmir Putin winning the next presidential election in Russia? While some might think the probability for that is 0.2, others might think it is 0.25. The Beta Distribution is a concept that provides a way of explaining this.

The Examples of Beta Distribution

The Beta Distribution can be used for representing the different probabilities as follows.

1. The likelihood of the audience rating the new movie release.

2. The click-through rate of the website, which is the proportion of visitors.

3. The conversion rate for buyers actually purchasing from your website.

4. What is the survival chance of a person having blood cancer.

The Formula for the Beta Distribution

The standard formula for Beta Distribution pdf is as follows. 

[f(x)=frac{(x−a)^{p−1}(b−x)^{q−1}}{B(p,q)(b−a)^{p+q−1}} ] a≤x≤b;p,q>0

Here, p and q represent the shape parameters. ‘A’ and ‘b’ are used for representing lower and the upper bounds respectively for the distribution. B(p, q) is the beta function. The beta function has this formula:

[ B(alpha,beta) = int_{1}^{0}t^{(α−1)}(1−t)^{(beta−1)}dt. ]

An event where the value of a = 0, and b = 1, is known as the standard Beta Distribution. Mathematical equation or formula related to standard Beta Distribution can be described as: [F (x) = frac{x^{p−1} (1−x)^{q−1}}{ B (p,q)} ] 0≤x≤1;p,q>0. 

Generally the usual form of the distribution is described with regards to scale and location parameters. The beta is a little different in the sense the usual distribution regarding upper and lower bounds is described. However, the scale and location parameters are specified in the form of lower and the upper limits as mentioned below. 

Location = a

Scale  = b – a

The Application of Beta Density Function

Beta Distribution is implemented and integrated in a wide range of applications like Bayesian hypothesis testing, task duration modelling, and Rule of Succession. The Beta Distribution is particularly the right project and planning control systems such as CPM and PERT primarily due to the fact that function is contrived by the interval with the max value of 1 and min value of 0. 

Solved Examples

There are various examples of Beta Distribution probability and solving them can help the students to understand them well and prepare for their exams. For instance you can find out about the probability of someone going out on a movie with you using the method of Beta Distribution. You can refer to the notes for numerous solved examples and explanations behind it.