Problem 3.1 Gamma function

Γ (x) = \int_0^{\infty} u^{x - 1} e^{- u}, d u

Integrate by parts:

\begin{align} \Gamma (x+1) & = \int\_{0}^{\infty} u^{x} e^{-u} , du \\ & = \int\_{0}^{\infty} u^{x} (-e^{-u})' d u \\ & = \left\[ u^{x}(-e^{-u}) \right]_{0}^{\infty} - \int_{0}^{\infty} (-e^{-u}) (x u ^{x-1}) , du \\ & = - \lim\_{ u \to \infty } \frac{u^{x}}{e^{u}} + x \int\_{0}^{\infty} e^{-u}u^{x-1} , du \\ & = x\Gamma(x) \end{align}

\Gamma(1) = \int\_{0}^{\infty} e^{-u} , du \stackrel{s(u)=-u}= - \int\_{0}^{-\infty} e^s , ds = - \left\[ e^{s} \right]\_{0}^{-\infty} = 1

Problem 3.2 MAP for the exponential density

let $x$ has an exponential density

p (x ∣ θ) = {θ e^{- θ x} 0,, x > 0 otherwise

the prior

p (x ∣ π) = π^{x} (1 - π)^{1 - x}

p (D ∣ π) = \prod_i = 1^{n} p (x_i ∣ π) = \prod_i = 1^{n} π^{x_i} (1 - π)^{1 - x_i} = π^{\sum_i = 1^{n} x_i} (1 - π)^{\sum_i = 1^{n} (1 - x_i)} = π^{s} (1 - π)^{1 - x}

p (π ∣ D) = \frac{p ( D ∣ π ) p ( π )}{\int p ( D ∣ π ) p ( π ) d π} =