The Beta Distribution is a continuous probability distribution defined on the interval [0, 1], widely used in statistics and various fields for modeling random variables that represent proportions or probabilities. It is particularly useful when dealing with scenarios where the outcomes are bounded within a specific range, such as success rates, probabilities, and proportions in Bayesian inference, decision theory, and reliability analysis. The flexibility of the Beta Distribution allows it to take on various shapes depending on its parameters, making it a versatile tool in mathematical modeling.
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).
Definition of the Beta Distribution
The Beta Distribution is parameterized by two positive shape parameters, α, and β, which determine the shape of the distribution.
The probability density function (PDF) of the Beta Distribution for a random variable X on the interval [0, 1] is given by:
f(x;α,β)= xα−1 (1−x)β−1 / B(α,β)
where 𝐵(𝛼,𝛽) is the Beta function, defined as:
B(α,β)=∫01 tα−1 (1−t)β−1 dt
This function serves as a normalization constant ensuring that the total probability over the interval [0, 1] equals 1.
Properties of the Beta Distribution
The Beta Distribution exhibits several key properties, which are essential for understanding its behavior and applications.
Probability Density Function (PDF)
The Probability Density Function (PDF) of the Beta Distribution describes the likelihood of a random variable X taking on a specific value within the interval [0, 1].
The shape of the PDF is influenced by the parameters α and 𝛽. Depending on these parameters, the distribution can be uniform, U-shaped, J-shaped, or bell-shaped, making it adaptable to various situations.
Probability Density FunctionCumulative Distribution Function (CDF)
The Cumulative Distribution Function (CDF) of the Beta Distribution, denoted as F(x;α,β), represents the probability that the random variable X is less than or equal to a specific value 𝑥.
It is computed as the integral of the PDF from 0 to 𝑥:
F(x;α,β)=∫0x f(t;α,β)dt
The CDF provides a way to calculate the probability of observing a value within a certain range, making it useful in hypothesis testing and confidence interval estimation.
Moments: Mean, Variance, Skewness, and Kurtosis
The moments of the Beta Distribution, including the mean, variance, skewness, and kurtosis, provide essential insights into the distribution's characteristics.
Mean:
The mean 𝐸[𝑋] of the Beta Distribution, which represents the central location or average value, is calculated as:
𝐸[𝑋]=𝛼/𝛼+𝛽
This formula shows that the mean is influenced by the relative values of the shape parameters 𝛼 and 𝛽. When 𝛼 equals 𝛽, the mean is 0.5, indicating a symmetric distribution centered around 0.5.
Variance:
The variance
Var(𝑋) measures the dispersion or spread of the distribution around the mean:
Var(𝑋)=𝛼𝛽 / (𝛼+𝛽)2(𝛼+𝛽+1)
A smaller variance indicates that the distribution is more concentrated around the mean, while a larger variance suggests a wider spread.
Skewness:
Skewness quantifies the asymmetry of the distribution. For the Beta Distribution, skewness is given by:
Skewness (𝑋)=2(𝛽−𝛼)√𝛼+𝛽+1/(𝛼+𝛽+2)√𝛼𝛽
- If 𝛼=𝛽, the distribution is symmetric (skewness = 0).
- If 𝛼>𝛽, the distribution is skewed to the left (negative skewness).
- If 𝛽>𝛼, the distribution is skewed to the right (positive skewness).
Kurtosis:
Kurtosis measures the "tailedness" or the concentration of data in the tails of the distribution:
Kurtosis(𝑋)=6[(𝛼−𝛽)2(𝛼+𝛽+1)𝛼𝛽(𝛼+𝛽+2)] / 𝛼𝛽(𝛼+𝛽+2)(𝛼+𝛽+3)
The kurtosis of the Beta Distribution can vary significantly based on the parameters 𝛼 and 𝛽, influencing the peak and tail behavior of the distribution.
Suppose an event can occur several times within a given unit of time. When the total number of occurrences of the event is unknown, we can think of it as a random variable. When a random variable X takes on values on the interval from 0 to 1, one choice of a probability density is the beta distribution whose probability density function is given by as follows.
Representation of probability density function -
f(x) = \frac{1}{\Beta(m.n)}x^{m-1}(1-x)^{n-1}
It will be applicable only when the given below given condition will pass.
x > 0, m >0, n >0.
f(x) = 0 , Otherwise
Here, you will see the meaning of function as you have shown in representation of probability density function where, B(m.n) is the value of the beta function.
Representation of B(m.n) -
\Beta(m,n) = \int^{1}_{0} x^{m- 1}(1-x)^{n-1}dx = \Beta(n,m)
Integrating it by parts, we will get the following expression as given below.
\Beta(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}
Where, Γ(x) is the gamma function of x, calculated as -
Γ(x) = (x-1)Γ(x-1) for x>1
=> \Gamma(x) = (x-1)! when x is an integer
The random variable X is represented as follows.
Representation of random variable X -
X ~ BETA(m,n)
Expected Value:
The Expected Value of the Beta distribution can be found by summing up products of Values with their respective probabilities.
\mu = E(X) = \int^{\infty}_{-\infty} x.f(x) dx
\mu = \frac{1}{\Beta(m,n)} \int^{1}_{0}x. x^{m - 1}(1-x)^{n-1} dx
\mu = \frac{1}{\Beta(m,n)} \int^{1}_{0} x^{m}(1-x)^{n-1} dx
Upon using the value of beta function, we will get the following expression a follows.
\mu = \frac{1}{\Beta(m,n)} \Beta(m+1,n)
but, \Beta(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}
also, \Beta(m+1,n) = \frac{\Gamma(m+1)\Gamma(n)}{\Gamma(m+n+1)}
=> \mu = \frac{\Gamma(m+1)\Gamma(n)}{\Gamma(m+n+1)} * \frac{\Gamma(m+n)}{\Gamma(m)\Gamma(n)}
Upon using the property of gamma function, that is Γ(x) = (x-1)!, we will get the following expression as follows.
So, E(x) = \frac{m! (n-1)! (m+n-1)!}{(m-1)! (n-1)! (m+n)!}
here, m, n -> integers
where provided m and n are integers.
Variance and Standard Deviation
The Variance of the Beta distribution can be found using the Variance Formula.
σ^2 = E( X − μ )^2 = E( X^2 ) − μ^2
E(X^2) = \int^{\infty}_{-\infty} x^2.f(x) dx
E(X^2) = \int^{1}_{0}\frac{1}{\Beta(m,n)}x^2. x^{m - 1}(1-x)^{n-1} dx
E(X^2) = \frac{1}{\Beta(m,n)}\int^{1}_{0}x^{m + 1}(1-x)^{n-1} dx
Upon using the value of beta function, we will get the following expression as follows.
E(X^2) = \frac{1}{\Beta(m,n)} \Beta(m+2,n+2)
but, \Beta(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}
also, \Beta(m+1,n) = \frac{\Gamma(m+2)\Gamma(n)}{\Gamma(m+n+2)}
=> \mu = \frac{\Gamma(m+2)\Gamma(n)}{\Gamma(m+n+2)} * \frac{\Gamma(m+n)}{\Gamma(m)\Gamma(n)}
Upon using the property of gamma function, that is Γ(x) = (x-1)!, we will get the following expression as follows.
E(X^2) = \frac{(m+1)!(n-1)!(m+n-1)!}{(m+n+1)!(m-1)!(n-1)!}
E(X^2) = \frac{m(m+1)}{(m+n+1)(m+n)}
So, Var(X) = E(X^2) - \mu^2
Var(X) = \frac{m(m+1)}{(m+n+1)(m+n)} - \frac{m^2}{(m+n)^2}
Var(X) = \sigma^2= \frac{m.n}{(m+n+1)(m+n)^2}
Standard Deviation is given by as follows.
\sigma = \sqrt{\frac{m.n}{(m+n+1)(m+n)^2}} = \frac{1}{(m+ n)}\sqrt{\frac{m.n}{(m+n+1)}}
Example -
In a certain county, the proportion of highway sections requiring repairs in any given year is a random variable having the beta distribution with m = 3 and n = 2.
(a) On the average, what percentage of the highway sections require repairs in any given year?
(b) Find the probability that at most half of the highway sections will require repairs in any given year.
Solution -
(a) \mu = \frac{3}{3+2} = 0.60
which means that on the average 60% of the highway sections require repairs in any given year.
(b) Substituting m=3 and n=2 in the probability density function, and
using Γ(3) = 2! = 2, Γ(2) = 1! = 1, Γ(5) = 4! = 24,
we will get the following expression as follows.
f(x) = 12 x^2 (1 − x) for 0<x<1. Otherwise f(x) = 0
Thus, the desired probability is given by as follows.
\int^{\frac{1}{2}}_0 12x^2 (1-x) dx = \frac{5}{16}
Shape of the Beta Distribution
The shape of the Beta Distribution is determined by the parameters
𝛼 and 𝛽, allowing it to take on various forms:
Uniform Distribution:
When 𝛼=𝛽=1, the Beta Distribution becomes a uniform distribution on the interval [0, 1], meaning every value within this interval is equally likely.
Bell-Shaped Distribution:
When 𝛼>1 and 𝛽>1, the Beta Distribution takes on a bell-shaped, unimodal form. The peak of the distribution occurs at 𝛼−1 / 𝛼+𝛽−2, making it suitable for modeling distributions with a central tendency.
U-Shaped Distribution:
When 𝛼<1 and 𝛽<1, the distribution is U-shaped, with most of the probability mass concentrated at the boundaries 0 and 1. This form is useful for modeling extreme probabilities.
J-Shaped Distribution:
When 𝛼<1 and 𝛽>1, the distribution is skewed towards 0, resembling a J-shape.
When 𝛼>1 and 𝛽<1, the distribution is skewed towards 1, also taking on a J-shaped form.
Applications of the Beta Distribution
The Beta Distribution's versatility makes it valuable across various fields:
Bayesian Inference:
The Beta Distribution is frequently used as a prior distribution in Bayesian analysis, especially for modeling binary outcomes and probabilities. Its conjugate nature with the binomial distribution simplifies posterior calculations.
Project Management (PERT):
In project management, the Beta Distribution is used in the Program Evaluation and Review Technique (PERT) to model the distribution of task completion times. It helps in estimating the expected duration and the likelihood of meeting deadlines.
Reliability Engineering:
The Beta Distribution is applied to model the life expectancy and failure rates of systems and components, particularly when data is sparse or when modeling expert opinions.
Genetics and Population Studies:
The Beta Distribution is employed to model the distribution of allele frequencies in populations, providing insights into genetic diversity and evolutionary dynamics.
Quality Control:
In quality control processes, the Beta Distribution is used to model the distribution of defect rates, helping organizations understand and manage variability in production processes.
The Beta Distribution's ability to model proportions, probabilities, and bounded outcomes makes it a powerful tool in both theoretical and applied statistics.
Solved Examples on Beta Distribution Model
Example 1: For a Beta Distribution with parameters α = 2 and β = 5:
- Mean: Calculate using the formula
α / (α + β). Thus, the mean is 2 / (2 + 5) = 2 / 7 ≈ 0.286. - Variance: Use the formula
αβ / [(α + β)² (α + β + 1)]. Therefore, the variance is 2 * 5 / [7² * 8] = 10 / 49 * 8 ≈ 0.0255.
Example 2: In a survey with α = 3 and β = 7, find the probability that the proportion is less than 0.4:
- Solution: To find this, use the cumulative distribution function (CDF) of the Beta Distribution evaluated at 0.4 with the given parameters.
Example 3: For α = 4 and β = 4, compute the skewness:
- Skewness: Using the formula
2 (β - α) sqrt(α + β + 1) / [(α + β + 2) sqrt(αβ)], the skewness is 2 * (4 - 4) * sqrt(8) / [(10) * sqrt(16)] = 0 because α equals β.
Example 4: With α = 1 and β = 2, find the probability that X is between 0.2 and 0.5:
- Solution: Calculate the difference between the CDF values at 0.5 and 0.2 using the given parameters.
Example 5: For α = 5 and β = 3, compute the kurtosis:
- Kurtosis: Substitute α and β into the kurtosis formula and simplify to obtain the value.
Practice Problems
- Compute the mean and variance for a Beta Distribution with α = 7 and β = 3.
- Find the probability that X is less than 0.6 for a Beta Distribution with α = 4 and β = 5.
- Determine the skewness of a Beta Distribution with α = 2 and β = 6.
- For α = 6 and β = 2, calculate the probability that X is greater than 0.3.
- Find the kurtosis for a Beta Distribution with α = 3 and β = 8.
- Compute the mean for a Beta Distribution where α = 10 and β = 15.
- Calculate the variance of a Beta Distribution with α = 5 and β = 5.
- Determine the CDF value at x = 0.7 for α = 2 and β = 3.
- Find the probability that X is between 0.4 and 0.8 for α = 4 and β = 6.
- Compute the skewness of a Beta Distribution with α = 8 and β = 4.
Summary / Conclusion
The Beta Distribution is a powerful and flexible tool used in statistics to model random variables constrained within the interval [0, 1]. Its versatility, governed by the parameters α and β, allows it to model various shapes of distributions, from uniform to bell-shaped, U-shaped, and J-shaped. The distribution's mean, variance, skewness, and kurtosis provide deep insights into its behavior, making it suitable for diverse applications such as Bayesian inference, project management, reliability engineering, genetics, and quality control. Understanding these properties helps in effective modeling and analysis of bounded outcomes.
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