ABSTRACT
Consider an event x that follows a normal distribution with mean μθ and variance σ2 where both μ and σ2 are known. The parameter θ follows a uniform distribution on the [0,1] interval. Thus, we may write
σ θθ π θ
πσ
(J.1)
Since we are interested in the posterior probability that θ is less than some value c where 0 ≤ c ≤ 1, we can write
(J.2)
σ θ πσ
= ∫ for 0 ≤ b ≤ 1, we can write
θ≤ ≤ =P (J.3)
Begin with
σ σθ θ πσ πσ
μ πσ
= ∫ (J.4)
= converting the variable w which is normally distributed
with mean x and variance σ2 to a standard normal variable. We write
where ( )zΦ is the cumulative standard normal distribution function. We can now substitute the right-hand side of (J.5) into (J.3) to write
P
the desired result.
