Question

Show that the graph of the \(\beta\) pdf is symmetric about the vertical line through \(x=\frac{1}{2}\) if \(\alpha=\beta\).

Short answer

When \(\alpha = \beta\), the Beta PDF, \(f(x; \alpha, \alpha)\), becomes \(f(1-x; \alpha, \alpha)\) after the substitution \(y = 1 - x\). Here we find that the functions are equivalent, hence the Beta PDF is symmetric about x=1/2.

Step-by-step solution

Understanding the Beta Function

The Probability density function (PDF) of the Beta distribution, for \(x \) in \([0,1]\), is given by \(f(x; \alpha, \beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)}\) where \(B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}\) is the Beta function and \(\Gamma(z)\) is the Gamma function.

Substitution of Variable

To prove symmetry, introduce the substitution \( y = 1 - x\). This allows to reflect the \(\alpha\) and \(\beta\) terms, transforming into \(f(y; \alpha, \beta) = \frac{y^{\alpha-1}(1-y)^{\beta-1}}{B(\alpha, \beta)}\).

Verifying Symmetry for Equal Alpha and Beta

Where alpha equals beta, the PDF simplifies to \(f(x; \alpha, \alpha) = \frac{x^{\alpha-1}(1-x)^{\alpha-1}}{B(\alpha, \alpha)}\). By substituting y = 1-x, it becomes \(f(1-x; \alpha, \alpha) = \frac{(1-x)^{\alpha-1}x^{\alpha-1}}{B(\alpha, \alpha)}\), which is equal to the original function. This proves that the Beta PDF is symmetric about x=1/2 when \(\alpha = \beta\).

Key concepts

Probability Density Function

A probability density function (PDF) is a crucial concept in statistics, vital for understanding various types of distributions, including the Beta distribution. In essence, a PDF describes the likelihood of a random variable to take on a given value. For continuous variables, the PDF provides the relative likelihood that the variable falls within a particular range of values, as opposed to discrete distributions that use a probability mass function (PMF).

To visualize this, imagine a curve on a graph where the total area under the curve represents 1, or 100% probability. At any given point on the x-axis, the height of the curve (the PDF value) gives the probability density of the variable at that point. For the Beta distribution, the PDF is particularly interesting because it can take on a variety of shapes depending on its parameters, which makes it useful for modelling phenomena that have bounded intervals, such as percentages and proportions.

At its core, the concept of PDF helps us understand the distribution and behavior of random variables within a defined range, making it a fundamental tool for probability and statistics.

Beta Function

The Beta function, denoted as \(B(\alpha, \beta)\), is an essential part of the Beta distribution's probability density function. Mathematically, it is related to the Gamma function and serves to normalize the PDF so that the total area under the curve is 1, ensuring it measures probabilities correctly.

The Beta function is defined as the integral of \(x^{\alpha-1}(1-x)^{\beta-1}\) over the interval from 0 to 1. This function shows symmetry when \(\alpha = \beta\), adding an interesting property to the Beta distribution’s PDF. When parameters \(\alpha\) and \(\beta\) are equal, the PDF becomes symmetric around the vertical line through \(x = \frac{1}{2}\), implying that for any given value of \(\alpha\), there is an equal chance of the variable falling on either side of \(\frac{1}{2}\).

The Beta function is a powerful tool in understanding how the shape of the Beta distribution changes with different parameter values, allowing for a deeper grasp of this versatile distribution used in various fields, such as finance, engineering, and social sciences.

Gamma Function

The Gamma function, denoted as \(\Gamma(z)\), plays an integral role in both the Beta function and the overall Beta distribution. It is a concept that extends the factorial function to real and complex numbers. If you are familiar with factorials, which are defined as the product of all positive integers up to a given number (e.g., \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\)), the Gamma function is the continuous counterpart for non-integer values.

The Gamma function is defined as \(\Gamma(z) = \int_0^{\infty} t^{z-1}e^{-t}dt\), for complex numbers with a real part greater than zero. One interesting property of \(\Gamma\) is that \(\Gamma(n) = (n-1)!\) for a positive integer \(n\). In the Beta PDF, the Gamma function is used to derive the Beta function because \(B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}\), and hence it directly impacts the shape and properties of the Beta distribution.

Understanding the Gamma function is vital because it serves as a building block for various functions and distributions in statistics and probability theory. Its connection to factorial allows a smooth extension into the realms of the continuous domain, bridging gaps between discrete and continuous analysis.