Consider the random variable (epsilon>0) such that [ Phi=ln left(frac{epsilon}{epsilon_{0}} ight) ] is Gaussian with mean (langlePhiangle)

Consider the random variable \(\epsilon>0\) such that

\[ \Phi=\ln \left(\frac{\epsilon}{\epsilon_{0}}\right) \]

is Gaussian with mean \(\langle\Phiangle\) and variance \(0<\sigma^{2}<\infty, 0<\epsilon_{0}<\infty\) is a reference value.

11.2.1 Compute the Pdf \(f(\epsilon)\) of \(\epsilon\).

11.2.2 Determine the moments \(\left\langle\epsilon^{n}\rightangle\) for \(n>0\) and integer as function of \(\langle\Phiangle, \sigma\) and \(n\).11.2.3 Set \(\epsilon_{0}=\langle\epsilonangle\) and compute the ratio \(\frac{\left\langle\epsilon^{n}\rightangle}{\langle\epsilonangle^{n}}\).

11.2.4 Let

\[ \frac{\left\langle\epsilon^{2}\rightangle}{\langle\epsilonangle^{2}}=A\left(\frac{L}{r}\right)^{\mu} \]

where \(A, L, \mu\) are positive constants, \(r>0\) is a parameter. Establish the dependence of \(\left\langle\epsilon^{n}\rightangle /\langle\epsilonangle^{n}\) on \(A, L / r, \mu\) and \(n\).