1. The Beta distribution: A Beta random variable is a positive continuous random variable
dened on [0; 1] that has two parameters associated with it, usually denoted and . Both
and must be positive real numbers. The beta distribution is used to model the probability
of success in Bernoulli trials when each trial has a random success probability - e.g. tossing
randomly selected coins.
The density function of a beta random variable X with parameters [itex]\alpha[/itex] and [itex]\beta[/itex] is
for 0 < x < 1.
The Beta distribution gets its name from the fact that its density function involves the so-called
Beta function. Here are several facts about the Beta function B(s; t):
(a) Let X be Beta with arbitrary parameters and . Show how to use the facts above
about the Beta function to derive c.
(b) Use the fact above about the Beta function and the facts from Homework 6 about the
Gamma function to find E(X) and E(X2)
) without actually doing any integration yourself.
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The Attempt at a Solution
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