I don't know the textbook used by MIT, but I bought a book from Dover Publications because I read a high recommendation for it on Amazon. The book is Partial Differential Equations For Scientists And Engineers by Stanley J. Farlow.
A random variable X has a binomial distribution with probability function
f(x|θ) = (nCx)θ^x × (1 − θ)^(n − x) where x = 1, 2, ... , n.
Nothing is known about the parameter θ, so a uniform (that is, vague) prior distribution
on the interval [0, 1] is assigned to θ. If 4 trials bring back 3 successes, then the posterior
probability density function of θ may be set down as:
π(θ|x) = f(x|θ)π(θ) ÷ ∫f(x|θ)π(θ)dθ, equal in this case to
(4C3)θ^3 × (1 − θ)^(4 − 3) ÷ ∫(from 0 to 1)(4C3)θ^3 × (1 − θ)dθ
which is proportional to θ^3(1 − θ).
θ^3(1 − θ) is seen for (α, β) = (4, 2) to conform to the beta density function
f(x) = x^(α − 1) × (1 − x)^(β − 1) ÷ B(α, β) for 0 < x < 1 & α, β > 0; f(x) is 0 otherwise.
The normalizing constant here should be 1 ÷ B(4, 2) which translates to 5! ÷3! or 20.
It then follows that the constant of proportionality is 20 and π(θ|x) is 20θ^3(1 − θ) for 0 < θ < 1.
