Question: Let Y denote astandardCauchyrandomvariable,thatis,a t-distributed randomvariablewith df = 1. (a) Fromthedefinitionofthe t distribution in Section 4.4.5, showthat Y can beexpressed as aratiooftwoindependentstandardnormalrandomvariables. (b) Simulate(i)1,(ii)100,(iii)10,000,(iv)100,000,(v)1,000,000,(vi)10,000,000standard Cauchyrandomvariables,eachtimefindingthesamplemean.Doesthesamplemean

Let Y denote astandardCauchyrandomvariable,thatis,a t-distributed randomvariablewith df = 1.

(a) Fromthedefinitionofthe t distribution in Section 4.4.5, showthat Y can beexpressed as aratiooftwoindependentstandardnormalrandomvariables.

(b) Simulate(i)1,(ii)100,(iii)10,000,(iv)100,000,(v)1,000,000,(vi)10,000,000standard Cauchyrandomvariables,eachtimefindingthesamplemean.Doesthesamplemean seem tobeconverging,asthelawoflargenumberspredicts?Whyorwhynot?Forthe n = 10, 000, 000 case, constructaboxplotandreportthefive-numbersummarytoshow howsomeobservationscanbeveryfarfromthequartilesof −1.0 and 1.0.Dothesample median andquartilesseemtoconvergetothetruevaluesas n increases?

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