Math Solutions

[Solution] (a) Let X˜N(μ ,σ ^2). Show that G_X(s)=E(s^X)=e^μ s+1/2σ ^2s^2 (b) Let X˜N(μ _1,σ _1^2) and Y˜N(μ

Question: (a) Let \(X\tilde{\ }N\left( \mu ,{{\sigma }^{2}} \right)\). Show that

\[{{G}_{X}}\left( s \right)=E\left( {{s}^{X}} \right)={{e}^{\mu s+\frac{1}{2}{{\sigma }^{2}}{{s}^{2}}}}\]

(b) Let \(X\tilde{\ }N\left( {{\mu }_{1}},\sigma _{1}^{2} \right)\) and \(Y\tilde{\ }N\left( {{\mu }_{2}},\sigma _{2}^{2} \right)\) be two independent normal random variables. Let Z = X + Y. Show that \(Z\tilde{\ }N\left( {{\mu }_{1}}+{{\mu }_{2}},\sigma _{1}^{2}+\sigma _{2}^{2} \right)\).

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