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Question Number 143593 by pticantor last updated on 16/Jun/21
$${soit}\:\left(\boldsymbol{{X}}_{{i}} \right),{i}\in\left\{\mathrm{1},\mathrm{2},......,{n}\right\}\:{une}\:{suite}\:{de}\:{variable}\:{a}\boldsymbol{{leartoire}}\:\boldsymbol{{independante}} \\ $$$$\left(\boldsymbol{{iid}}\right)\:\boldsymbol{{suivant}}\:\boldsymbol{{la}}\:\boldsymbol{{loi}}\:\boldsymbol{{binomiale}}\:\boldsymbol{{B}}\left(\boldsymbol{{n}},\boldsymbol{{p}}\right)\: \\ $$$$\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\overset{\_} {\boldsymbol{{X}}}_{\boldsymbol{{n}}} \boldsymbol{{converge}}\:\boldsymbol{{en}}\:\boldsymbol{{loi}}\:\boldsymbol{{vers}}\:\boldsymbol{{E}}\left(\boldsymbol{{X}}\right) \\ $$$$\mathrm{2}−\:\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}}\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}} {\sum}}\boldsymbol{{X}}_{\boldsymbol{{i}}\:\:} ^{\mathrm{2}} \boldsymbol{{converge}}\:\boldsymbol{{en}}\:\boldsymbol{{loi}}\:\:\boldsymbol{{vers}}\:\:\:\:\:\:\:\:\boldsymbol{{E}}\:\left(\boldsymbol{{X}}^{\mathrm{2}} \right) \\ $$$$\mathrm{3}−\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\boldsymbol{{X}}_{{i}} \boldsymbol{{Y}}_{{i}} \:\boldsymbol{{converge}}\:\boldsymbol{{en}}\:\:\boldsymbol{{loi}}\:\boldsymbol{{vers}}\:\boldsymbol{{E}}\left(\boldsymbol{{XY}}\right) \\ $$