Question

Let 𝑋 be a CRV with PDF

                       

a. Find the value of the constant 𝑐.

b. Show that 𝜃̂ = 3𝑋̅ is an unbiased estimator for 𝜃.

c. Find the maximum likelihood estimator for 𝜃.

 

Answer 1

Solution

Let 𝑋 be a CRV with PDF

                       

a. Find the value of the constant 𝒄.

We have

                       

Therefore, c=1/2

b. Show that 𝜽̂=𝟑𝑿̅ is an unbiased estimator for 𝜽.

we have 𝐸(𝜃̂)=𝐸(3𝑋̅)=3𝐸(𝑋̅)

                     

Therefore: 𝜽̂=𝟑𝑿̅ 𝐢𝐬 𝐚𝐧 𝐮𝐧𝐛𝐢𝐚𝐬𝐞𝐝 𝐞𝐬𝐭𝐢𝐦𝐚𝐭𝐨𝐫 𝐟𝐨𝐫 𝛉

c. Find the maximum likelihood estimator for 𝜽.

We got:

                             

Then:

                             

We got:

                             

Therefore,,

                                       

Then we have :