Beta Distribution: Definition, Formulas, Examples, Properties

Beta Distribution, in the probability theory, can be described as a continuous probability distribution family. It is defined on the basis of the interval [0, 1]. The concept of Beta distribution also represents the value of probability. Generally, this is a basic statistical concept. For example, you have to finish a complicated task. Now, even if you plan all the necessary steps beforehand, some issues may come up without prior notice and this might hamper your calculated time to finish the work, right? Hence, there you might need to calculate the probability of the time to finish that job. And here comes the concept of the Beta distribution. Here, we will be discussing more about beta distribution along with some important questions. 

Key takeaways: Beta distribution, probability, probability distribution, beta distribution notation, beta distribution formula.

Also Read: Relations and Functions

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Definition of Beta Distribution

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The Concept of Beta distribution can be defined as the family of continuous probability distribution which is defined on the basis of intervals [0, 1]. It has two positively shaped parameters - α and β, which represents the exponents of a random variable and manage the formation of the distribution.

This process of Beta distribution can be divided into two kinds- the basic distributions are called as Beta distribution of the First Kind, whereas the prime ones are called the beta distribution of the Second Kind.

Also Read: Differentiation and Integration Formula

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Examples of Beta Distribution

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This concept is applicable to a lot of real-life situations, for example-

1. For calculating the expected customer reviews after launching new products.
2. To calculate the online visitors shopping activities of the online shopping sites.
3. To understand and analyze the survival chances of the patients suffering from particular health conditions.
4. The probable time for completing a task.

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Beta Distribution Notation

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We know from the definition of Beta distribution, it is based on the interval [0, 1], and is represented by α and β. Now, these are counted as the two positive perimeters that are the exponents of the random variables and manage the formation of the distribution. Therefore, its notation is Beta, which includes α and β and this α and β are basically two real numbers.

Let’s have a look to the following picture to understand better.

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Beta Distribution Formulas

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The behaviour of random variables, limited to the intervals of a particularly measured length in a variety of disciplines, can be measured by the Beta distribution. Hence, we need to understand a few formulas while applying the Beta distribution concept to practical use.

Probability Density Function

Cumulative Density Function

CDF = Bx (α, β)/ B (α, β)

Where,

Moment Generating Function

Sometimes, Beta can be considered as the Moment Generating Function, and in that case-

Expectations

When it comes to the Expected Value of Beta distribution,

Hence, the Beta Variance is-

Also Read:

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Properties of Beta Distribution

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Here we will learn about some properties that satisfy the concept of Beta distribution, for example-

While measuring the Central tendency,

1. Mean
2. Median
3. Mode
4. Geometric Mean
5. Harmonic Mean

While calculating the statistical dispersion, like-

1. Variance
2. Geometric variance and covariance
3. Mean absolute difference
4. Mean absolute deviation around the mean

Also Read: Mean, Median and Mode

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Things to Remember

• Beta distribution is defined as the family of continuous probability distribution which is defined on the basis of the interval [0, 1].
• It includes two positively shaped parameters - α and β.
• It is basically a statistical concept of probability.
• The beta distribution is divided into two kinds- the Beta distribution of First Kind, and Beta distribution of the Second Kind.
• Formulas to Remember: 

Also Read: Integration by Partial Fractions