Solution Found!
Suppose that Y has a beta distribution with parameters ?
Chapter 4, Problem 200SE(choose chapter or problem)
Suppose that Y has a beta distribution with parameters \(\alpha\) and \(\beta\).
a If 𝑎 is any positive or negative value such that \(\alpha+a>0\), show that
\(E\left(Y^{a}\right)=\frac{\Gamma(\alpha+a) \Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\alpha+\beta+a)}\).
b Why did your answer in part (a) require that \(\alpha+a>0\)?
c Show that, with \(a=1\), the result in part (a) gives \(E(Y)=\alpha /(\alpha+\beta)\).
d Use the result in part (a) to give an expression for \(E(\sqrt{Y})\). What do you need to assume about \(\alpha\)?
e Use the result in part (a) to give an expression for \(E(1 / Y)\), \(E(1 / \sqrt{Y})\), and \(E\left(1 / Y^{2}\right)\). What do you need to assume about \(\alpha\) in each case?
Equation Transcription:
Text Transcription:
Alpha
beta
alpha.+a>0
E(Y^a)=Gamma(alpha+a)Gamma(alpha+beta) over Gamma(alpha)(alpha+beta+a)
alpha+a>0
a=1
E(Y)=alpha/(alpha+beta)
E(sqrt Y)
alpha
E(1/Y)
E(1/sqrt Y)
E(1/Y^2)
Alpha
Questions & Answers
QUESTION:
Suppose that Y has a beta distribution with parameters \(\alpha\) and \(\beta\).
a If 𝑎 is any positive or negative value such that \(\alpha+a>0\), show that
\(E\left(Y^{a}\right)=\frac{\Gamma(\alpha+a) \Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\alpha+\beta+a)}\).
b Why did your answer in part (a) require that \(\alpha+a>0\)?
c Show that, with \(a=1\), the result in part (a) gives \(E(Y)=\alpha /(\alpha+\beta)\).
d Use the result in part (a) to give an expression for \(E(\sqrt{Y})\). What do you need to assume about \(\alpha\)?
e Use the result in part (a) to give an expression for \(E(1 / Y)\), \(E(1 / \sqrt{Y})\), and \(E\left(1 / Y^{2}\right)\). What do you need to assume about \(\alpha\) in each case?
Equation Transcription:
Text Transcription:
Alpha
beta
alpha.+a>0
E(Y^a)=Gamma(alpha+a)Gamma(alpha+beta) over Gamma(alpha)(alpha+beta+a)
alpha+a>0
a=1
E(Y)=alpha/(alpha+beta)
E(sqrt Y)
alpha
E(1/Y)
E(1/sqrt Y)
E(1/Y^2)
Alpha
ANSWER:
Answer:
Step 1 of 5:
(a)
Suppose that has a beta distribution with parameters
If is any positive or negative value such that
show that
A random variable is said to have a beta probability distribution with parameters
if and only if the density function of
is
…………(1)
Where,
…………..(2)
We know the expected value of any random variable,
Hence,
[ is defined in the range of
]
[using equation (2)]
……(3)
We know from equation (2),
Hence compare the integral of equation (2) and (3), our parameter will become
………..(4)
Hence proved.