CT6 Q&A Part 1

[Bharti Singla]

Hii all
In Q&A Part-1 , Qus. 1.12- Here, in its (ii) part, we are told to find the Bayesian estimator under all-or-nothing loss. Well, we can easily find the mode of Beta distribution which is the posterior distribution here. But, if we are asked to find the estimator under 'Absolute error loss' , then how we can find the median of Beta distribution? Is there any easy method to find the median of Beta distribution and can it be asked in exam?
Thanks

 

[Varsha Agarwal]

You can integrate the pdf of Beta within limits 0<x<M,where M is the Median.
And then equate it to 0.5.

P.S : Anything can come in IAI.

 

[Bharti Singla]

But I am unable to integrate its pdf. Here the pdf is P^17 (1-P)^9 , so it would be very messy.

 

[Varsha Agarwal]

Integrate the original form of pdf of Beta and then put values.

 

[Bharti Singla]

Integrate the original form of pdf of Beta and then put values.

Hey, have you tried it? Still I'm unable to integrate its original pdf. Even its CDF is not given in tables.

 

[Varsha Agarwal]

Hey, have you tried it? Still I'm unable to integrate its original pdf. Even its CDF is not given in tables.

You cannot write inverse of CDF in this case..interpolate the value of M for which value of integrate pdf between 0<x<M is 0.5.

 

[John Lee]

The median of a Beta posterior has not been asked in the IFoA exams.

The particular case you have given would require repeated integration by parts and so is unlikely to be asked. However if the powers were lower then it would be feasible to integrate it. Then you would have to use numerical methods to solve for M.