Statistics (3rd Year) - Special Topic (Bayesian statistical methods)

Question

Suppose that the proportion of defective items θ in a large manufactured lot is unknown and the prior distribution of θ is Uniform(0,1) distribution. Items are selected at random from the lot one by one until exactly four defective are found. A total of 10 items were selected and tested. What is the posterior distribution of θ? Hint: Use Geometric distribution.

Answer 1

The prior is f(θ) = 1 0 <= θ <= 1

Then

With continuous distributions, f(θ|x) = f(x|θ)f(θ)/ I(-∞,∞)f(θ) f(x|θ)dθ

This is actually the negative binomial. The probability that 10 items were tested to get 4 failures is

C(9,3)θ⁴(1-θ)^10-4 = C(9,3)θ⁴(1-θ)⁶

Then, we have

f(θ|x) = f(x|θ)f(θ)/ I(-∞,∞)f(θ) f(x|θ)dθ =

C(9,3)θ⁴(1-θ)⁶ /I[0,1]C(9,3)θ⁴(1-θ)⁶ dθ =

θ⁴(1-θ)⁶ /I[0,1]θ⁴(1-θ)⁶ dθ

We can solve this either by integrating the denominator (integration by parts 4 times) or noting that the distribution will be

Cθ⁴(1-θ)⁶ from 0 to 1, which is B(4+1, 6+1) = B(5, 7)

C = Γ(5 + 7)/(Γ(5)Γ(7)) = 11!/(4!6!) = 11 * C(10,4) = 11 * 210 = 2310

f(θ|x) = Γ(5 + 7)/(Γ(5)Γ(7)) = 2310 θ⁴(1-θ)⁶ or simply f(θ|x) ~ B(5,7)