In Chapter 6 we will see that if Y is beta distributed

QUESTION:

Problem 120E

In Chapter 6 we will see that if Y is beta distributed with parameters α and β, then Y ∗ = 1 – Y has a beta distribution with parameters α ∗ = β and β ∗ = α. Does this explain the differences and similarities in the graphs of the beta densities in Exercises 4.118 and 4.119?

Reference

Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (α = 4, β = 0.3), (α = 7, β = 0.3), and (α = 12, β = 0.3).

a Are these densities symmetric? Skewed left? Skewed right?

b What do you observe as the value of α gets closer to 12?

c Which of these beta distributions gives the highest probability of observing a value less than 0.8?

d Graph some more beta densities withα > 1 andβ < 1. What do you conjecture about the shape of beta densities with α > 1 andβ < 1?

Applet Exercise Use the applet Comparison of Beta Density Functions to compare beta density functions with (α = .3, β = 4), (α = .3, β = 7), and (α = .3, β = 12).

a Are these densities symmetric? Skewed left? Skewed right?

b What do you observe as the value of β gets closer to 12?

c Which of these beta distributions gives the highest probability of observing a value larger than 0.2?

d Graph some more beta densities withα < 1 andβ > 1. What do you conjecture about the shape of beta densities with α < 1 andβ > 1?