Let X have a beta distribution with parameters ? and ?.

Chapter 5, Problem 8E

(choose chapter or problem)

Let \(X\) have a beta distribution with parameters \(\alpha\) and \(\beta\). (See Example 5.2-3.)

(a) Show that the mean and variance of \(X\) are, respectively,

\(\mu=\frac{\alpha}{\alpha+\beta} \quad \text { and } \quad \sigma^{2}=\frac{\alpha \beta}{(\alpha+\beta+1)(\alpha+\beta)^{2}}\) .

(b) Show that when \(\alpha>1\) and \(\beta>1\), the mode is at \(x=(\alpha-1) /(\alpha+\beta-2)\).

Equation Transcription:

 

 


 

Text Transcription:

X

Alpha

Beta

mu =alpha/alpha + beta

sigma^2= alpha beta /(alpha + beta +1)(alpha+ beta)^2  

alpha>1  

Beta >1  

x=(alpha-1)/(beta+-2)

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