P ( Y = − 192 ) = P ( X = − 4 ) = 1 5 P ( Y = − 3 ) = P ( X = − 1 ) = 1 5 P ( Y = 24 ) = P ( X = 2 ) = 1 5 P ( Y = 81 ) = P ( X = 3 ) = 1 5 P ( Y = 192 ) = P ( X = 4 ) = 1 5 f Y ( y ) = { 0.2 , y ∈ { − 192 , − 3 , 24 , 81 , 192 } 0 , o t h e r w i s e F Y ( y ) = { 0 , y < − 192 0.2 , − 192 ⩽ y < − 3 0.4 , − 3 ⩽ y < 24 0.6 , 24 ⩽ y < 81 0.8 , 81 ⩽ y < 192 1 , y ⩾ 192 E Y = − 192 − 3 + 24 + 81 + 192 5 = 20.4 E Y 2 = ( − 192 − 20.4 ) 2 + ( − 3 − 20.4 ) 2 + ( 24 − 20.4 ) 2 + ( 81 − 20.4 ) 2 + ( 192 − 20.4 ) 2 5 = 15758.6 v a r ( Y ) = 15758.6 − 20. 4 2 = 15342.5 P\left( Y=-192 \right) =P\left( X=-4 \right) =\frac{1}{5}\\P\left( Y=-3 \right) =P\left( X=-1 \right) =\frac{1}{5}\\P\left( Y=24 \right) =P\left( X=2 \right) =\frac{1}{5}\\P\left( Y=81 \right) =P\left( X=3 \right) =\frac{1}{5}\\P\left( Y=192 \right) =P\left( X=4 \right) =\frac{1}{5}\\f_Y\left( y \right) =\left\{ \begin{array}{c} 0.2,y\in \left\{ -192,-3,24,81,192 \right\}\\ 0,otherwise\\\end{array} \right. \\F_Y\left( y \right) =\left\{ \begin{array}{c} 0,y<-192\\ 0.2,-192\leqslant y<-3\\ 0.4,-3\leqslant y<24\\ 0.6,24\leqslant y<81\\ 0.8,81\leqslant y<192\\ 1,y\geqslant 192\\\end{array} \right. \\EY=\frac{-192-3+24+81+192}{5}=20.4\\EY^2=\frac{\left( -192-20.4 \right) ^2+\left( -3-20.4 \right) ^2+\left( 24-20.4 \right) ^2+\left( 81-20.4 \right) ^2+\left( 192-20.4 \right) ^2}{5}=15758.6\\var\left( Y \right) =15758.6-20.4^2=15342.5 P ( Y = − 192 ) = P ( X = − 4 ) = 5 1 P ( Y = − 3 ) = P ( X = − 1 ) = 5 1 P ( Y = 24 ) = P ( X = 2 ) = 5 1 P ( Y = 81 ) = P ( X = 3 ) = 5 1 P ( Y = 192 ) = P ( X = 4 ) = 5 1 f Y ( y ) = { 0.2 , y ∈ { − 192 , − 3 , 24 , 81 , 192 } 0 , o t h er w i se F Y ( y ) = ⎩ ⎨ ⎧ 0 , y < − 192 0.2 , − 192 ⩽ y < − 3 0.4 , − 3 ⩽ y < 24 0.6 , 24 ⩽ y < 81 0.8 , 81 ⩽ y < 192 1 , y ⩾ 192 E Y = 5 − 192 − 3 + 24 + 81 + 192 = 20.4 E Y 2 = 5 ( − 192 − 20.4 ) 2 + ( − 3 − 20.4 ) 2 + ( 24 − 20.4 ) 2 + ( 81 − 20.4 ) 2 + ( 192 − 20.4 ) 2 = 15758.6 v a r ( Y ) = 15758.6 − 20. 4 2 = 15342.5
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