CHEBYSHEV’S INEQUALITY
Let "X" be a discrete rv with mean "\\mu" and standard deviation "\\sigma". Then, for any "k\\geq1,"
Given that "\\mu=64.5, \\sigma^2=144"
"a)\\ \\ P(44<X<85)"
"{85-44 \\over 2}=20.5"
"k={20.5 \\over 12}"
"P(44<X<85)=1-P(|X-\\mu|\\geq k\\sigma)="
"=1-P\\bigg(|X-64.5|\\geq {20.5 \\over 12}\\cdot 12\\bigg)\\geq1-{1 \\over (20.5\/12)^2}"
"P(44<X<85)>0.6573"
"b) P(36<X<93)"
"{93-36 \\over 2}=28.5"
"k={28.5 \\over 12}"
"P(36<X<93)=1-P(|X-\\mu|\\geq k\\sigma)="
"=1-P\\bigg(|X-64.5|\\geq {28.5 \\over 12}\\cdot 12\\bigg)\\geq1-{1 \\over (28.5\/12)^2}"
"P(36<X<93)>0.8227"
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