Problem 200SE

Suppose that Y has a beta distribution with parameters α and β.

a If a is any positive or negative value such that α + a > 0, show that

b Why did your answer in part (a) require that α + a > 0?

c Show that, with a = 1, the result in part (a) gives E(Y ) = α/(α + β).

d Use the result in part (a) to give an expression for E( √ Y ). What do you need to assume about α?

e Use the result in part (a) to give an expression for E(1/Y ), E(1/ √ Y ), and E(1/Y 2). What do you need to assume about α in each case?

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.