Actuarial Outpost Mahler Exam #4 Q12
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#1
12-13-2017, 08:32 AM
 Bash Member SOA Join Date: Jun 2010 Posts: 60
Mahler Exam #4 Q12

1. I am trying to solve Mahler's Exam 4 Q12 using conjugate prior:
The number of boys in a family with m children is Binomial. The parameter q varies between different families with Beta distribution where a=3, b=1, Θ=1.
What is the probability that a family with 6 children have 5 boys and 1 girl.

2. From the C table, Distribution for Beta is given as F(x) = Beta(a, b; u).
Can one evaluate F(x) without integrating? If yes how?

3. Is there any relationship between Beta and another distribution when a=b?

Thanks.
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#2
12-13-2017, 09:04 AM
 edcars CAS AAA Join Date: Jan 2016 College: UofM Posts: 25

The key to this problem is that Θ=1. Once you know this, the problem simplifies to (you can plug it into the distribution to find it out for yourself):

Γ(a+b)/Γ(a)Γ(b) x^(a-1) (1-x)^(b-1)

I found that they may also put the distribution in this form and you need identify a, b, and Θ from this. Hope this helps!
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#3
12-13-2017, 06:28 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,672

Quote:
 Originally Posted by Bash 1. I am trying to solve Mahler's Exam 4 Q12 using conjugate prior: The number of boys in a family with m children is Binomial. The parameter q varies between different families with Beta distribution where a=3, b=1, Θ=1. What is the probability that a family with 6 children have 5 boys and 1 girl. 2. From the C table, Distribution for Beta is given as F(x) = Beta(a, b; u). Can one evaluate F(x) without integrating? If yes how? 3. Is there any relationship between Beta and another distribution when a=b? Thanks.
I don't understand why you think you need to know that. This is just a straightforward mixture problem. If you pretend that you know q, the probability is 6 q^5 - 6 q^6. You then take the expected value of that as q varies, and the needed moments of the Beta are in the table.
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#4
12-20-2017, 10:25 AM
 Bash Member SOA Join Date: Jun 2010 Posts: 60

Thanks to Edcars and Jim.
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