stat609-26 - Lecture 26 Sums based on a random sample Sums...

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beamer-tu-logLecture 26: Sums based on a random sampleSums formed from a random sample are useful statistics.We now study their properties.Lemma 5.2.5.LetX1,...,Xnbe a random sample from a population and letg(x)be afunction such thatE[g(X1)and Var(g(X1))exist. Then,EbracketleftBiggni=1g(Xi)bracketrightBigg=nE[g(X1)]andVarparenleftBiggni=1g(Xi)parenrightBigg=nVar(g(X1))Proof.SinceX1,...,Xnhave the same distribution,E[g(Xi)] =E[g(X1)]for alliand, hence,EbracketleftBiggni=1g(Xi)bracketrightBigg=ni=1E[g(Xi)] =ni=1E[g(X1)] =nE[g(X1)]By definition,UW-Madison (Statistics)Stat 609 Lecture 2620141 / 11
beamer-tu-logLecture 26: Sums based on a random sampleSums formed from a random sample are useful statistics.We now study their properties.Lemma 5.2.5.LetX1,...,Xnbe a random sample from a population and letg(x)be afunction such thatE[g(X1)and Var(g(X1))exist. Then,EbracketleftBiggni=1g(Xi)bracketrightBigg=nE[g(X1)]andVarparenleftBiggni=1g(Xi)parenrightBigg=nVar(g(X1))Proof.SinceX1,...,Xnhave the same distribution,E[g(Xi)] =E[g(X1)]for alliand, hence,EbracketleftBiggni=1g(Xi)bracketrightBigg=ni=1E[g(Xi)] =ni=1E[g(X1)] =nE[g(X1)]By definition,UW-Madison (Statistics)Stat 609 Lecture 2620141 / 11
beamer-tu-logVarparenleftBiggni=1g(Xi)parenrightBigg=EbraceleftBiggni=1g(Xi)EbracketleftBiggni=1g(Xi)bracketrightBiggbracerightBigg2=EbraceleftBiggni=1{g(Xi)E[g(Xi)]}bracerightBigg2=EbraceleftBiggni=1{g(Xi)E[g(Xi)]}2bracerightBigg+EbraceleftBigginegationslash=j{g(Xi)E[g(Xi)]}{g(Xj)E[g(Xj)]}bracerightBigg=nE[g(X1)Eg(X1)]2=nVar(g(X1))Theorem 5.2.6.LetX1,...,Xnbe a random sample from a populationFonRwith meanμand varianceσ2. Thena.E(¯X) =μ;b. Var(¯X) =σ2/n;c.E(S2) =σ2.UW-Madison (Statistics)Stat 609 Lecture 2620142 / 11
beamer-tu-logVarparenleftBiggni=1g(Xi)parenrightBigg=EbraceleftBiggni=1g(Xi)EbracketleftBiggni=1g(Xi)bracketrightBiggbracerightBigg2=EbraceleftBiggni=1{g(Xi)E[g(Xi)]}bracerightBigg2=EbraceleftBiggni=1{g(Xi)E[g(Xi)]}2bracerightBigg+EbraceleftBigginegationslash=j{g(Xi)E[g(Xi)]}{g(Xj)E[g(Xj)]}bracerightBigg=nE[g(X1)Eg(X1)]2=nVar(g(X1))Theorem 5.2.6.LetX1,...,Xnbe a random sample from a populationFonRwith meanμand varianceσ2. Thena.E(¯X) =μ;b. Var(¯X) =σ2/n;c.E(S2) =σ2.UW-Madison (Statistics)Stat 609 Lecture 2620142 / 11

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