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2.(a)Let X be a positive continuous random variable with Vk∈N:EXk]∈R.

By December 6, 2021No Comments

2.(a)Let X be a positive continuous random variable with Vk∈N:EXk]∈R.i.Show that E[X]=(1-Fx(x))dax.Hint:Write 1-Fx(x)in terms ofan integral of the pdf,then switch the order of integration.[3]i.Find an expression for E[X](where k∈N)of the form J0(l-Fx(x)h(x)dxfor a function h which you should specify.[5](b)Now let Fx(x)=1-(1+x)-mifx≥00otherwise,where m∈N.i For which k∈N does the expectation EXk]exist?Justify your answer.[3]ii.Compute the cdf of the sample minimum X(1)of a random sample of sizen∈N from Fx.For which k∈N does E[(X()]exist?[4]

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