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Question about Beta distribution (probability)

  • Thread starter iiiiaann
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  • #1
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Homework Statement



1. The Beta distribution: A Beta random variable is a positive continuous random variable
de ned 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
Image_2011.10.21_2_57_51_AM.png

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):
Image_2011.10.21_2_59_00_AM.png

(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 fi nd E(X) and E(X2)
) without actually doing any integration yourself.
Check your answers against Wikipedia

Homework Equations



don't know of any

The Attempt at a Solution



don't know where to start either of these
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



1. The Beta distribution: A Beta random variable is a positive continuous random variable
de ned 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
Image_2011.10.21_2_57_51_AM.png

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):
Image_2011.10.21_2_59_00_AM.png

(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 fi nd E(X) and E(X2)
) without actually doing any integration yourself.
Check your answers against Wikipedia

Homework Equations



don't know of any

The Attempt at a Solution



don't know where to start either of these
What must be the value of 'c' in order that f(x) be a legitimate pdf? What is the formula for EX, in terms of the pdf f(x)? How can you express the result in terms of functions that you are permitted to use in this exercise (i.e., Beta and Gamma functions)? Ditto for E(X^2).

RGV
 
  • #3
22
0
i know that for f(x) to be a legitimate pdf, the integral from negative infinity to positive infinity of f(x) has to equal 1, but i don't know how that could help me find c in this situation
 
  • #4
30
0
But in this case, you don't need to integral it from -infinity to infinity. Only integral it over its domain (or over the interval where f is positive). Once you notice its domain, compare your resulting integral with the integral formula of the beta function. You should get the answer then.
 

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