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Geometric Random variables ใ€‚Fork>1, P(X=kx >1)= P(X=k 2=_(1-p)k-1p (1-p)k-1p P(X>1) 280-2(1-p)k-1p -p(1-p)1-4-p 1 =(1-p)k-2p=px(k-1) ·EXx>1]=โˆ‘K=2kg(X=kX>1) -โˆ‘px-1)=โˆ‘k+1prไพง=โˆ‘px+โˆ‘xไธช =ๅญE[X] k=1 k=1 3Geometric Random Variables โ€ข For ๐‘˜ > 1, ๐‘ƒ ๐‘‹ = ๐‘˜ ๐‘‹ > 1 = ๐‘ƒ(๐‘‹=๐‘˜) ๐‘ƒ(๐‘‹>1) = (1โˆ’๐‘) ๐‘˜โˆ’1๐‘ ฯƒ๐‘˜=2 โˆž (1โˆ’๐‘) ๐‘˜โˆ’1๐‘ = (1โˆ’๐‘) ๐‘˜โˆ’1๐‘ ๐‘(1โˆ’๐‘)โˆ™ 1 1โˆ’(1โˆ’๐‘) = (1 โˆ’ ๐‘) ๐‘˜โˆ’2๐‘ = ๐‘๐‘‹(๐‘˜ โˆ’ 1) โ€ข ๐ธ ๐‘‹ ๐‘‹ > 1 = ฯƒ๐‘˜=2 โˆž ๐‘˜๐‘ƒ ๐‘‹ = ๐‘˜ ๐‘‹ > 1 = เท ๐‘˜=2 โˆž ๐‘˜๐‘๐‘‹(๐‘˜ โˆ’ 1) = เท ๐‘˜=1 โˆž (๐‘˜ + 1)๐‘๐‘‹(๐‘˜) = เท ๐‘˜=1 โˆž ๐‘˜๐‘๐‘‹(๐‘˜) + เท ๐‘˜=1 โˆž ๐‘๐‘‹(๐‘˜) = 1 + ๐ธ[๐‘‹] 3
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