Problem 2.5.1. Show that \(\displaystyle \int ^{\infty }_{-\infty }e^{-a x^{2}}dx = \sqrt {\pi /a}\) for \(a>0\), and deduce \(\Gamma (1/2)=\sqrt {\pi }\). Where to start: substitute \(u = x\sqrt {a}\) and compare with the definition of the error function at infinity.

Problem 2.5.2. Prove that \(\operatorname {erfc}(x)\sim \dfrac {e^{-x^{2}}}{x\sqrt {\pi }}\) as \(x\to \infty \). Where to start: integrate by parts once in \(\int ^{\infty }_{x}e^{-t^{2}}dt\), writing the integrand as \(\frac {1}{2t}\cdot 2t\,e^{-t^{2}}\), and show the remaining integral is of smaller order.

Problem 2.5.3. Express \(P(|Z|\leq 1.96)\) for \(Z\sim N(0,1)\) in terms of the error function, and verify numerically that it is approximately \(0.95\).

Problem 2.5.4. Using the recurrence for the incomplete gamma function, establish the identity \[\Gamma (n,x) = (n-1)!\,e^{-x}\sum ^{n-1}_{k=0}\frac {x^{k}}{k!}\] for positive integers \(n\), and interpret it in terms of a Poisson process.

Problem 2.5.5. Show that \(I_x(a,b) = 1 - I_{1-x}(b,a)\), and explain how this symmetry halves the work of tabulating the beta distribution. Where to start: substitute \(t\mapsto 1-t\) in the defining integral.

Problem 2.5.6. Show that for a \(\operatorname {Beta}(a,b)\) random variable \[E(X) = \frac {a}{a+b},\qquad \operatorname {var}(X) = \frac {ab}{(a+b)^{2}(a+b+1)},\] using the beta–gamma relation of Section 2.2 rather than integrating directly. Where to start: \(E(X^{k}) = \beta (a+k,b)/\beta (a,b)\), then apply \(\Gamma (z+1)=z\Gamma (z)\) repeatedly.