1) Let X be a random variable that takes values μ−a, μ, and μ+a with probabilities p, 1 − 2p, and p respectively. Show Pr(|X − μ| >= a) = Var(X)/a^2 2) For a discrete random variable X, show that m = E(X) minimizes E[(X − m)^2]
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Originally Posted by ashimb9 1) Let X be a random variable that takes values μ−a, μ, and μ+a with probabilities p, 1 − 2p, and p respectively. Show Pr(|X − μ| >= a) = Var(X)/a^2 Since \ |x-\mu| \ge a[/tex] when $\displaystyle x=\mu-a$ and $\displaystyle x=\mu+a$ (assuming $\displaystyle a>0$) $\displaystyle Pr(|x-\mu| \ge a)=2p $ You should know how to calculate $\displaystyle Var(X)$ to complete the demonstration. RonL
Originally Posted by CaptainBlack Since \ |x-\mu| \ge a[/tex] when $\displaystyle x=\mu-a$ and $\displaystyle x=\mu+a$ (assuming $\displaystyle a>0$) $\displaystyle Pr(|x-\mu| \ge a)=2p $ You should know how to calculate $\displaystyle Var(X)$ to complete the demonstration. RonL But how can it be assumed for certain that a>0 ?
Originally Posted by ashimb9 But how can it be assumed for certain that a>0 ? If $\displaystyle a<0$, then it is always true that $\displaystyle |x-\mu|>a$ since $\displaystyle |x-\mu|$ is always positive.
Originally Posted by ashimb9 But how can it be assumed for certain that a>0 ? You will probabbly find it as an extra condition somewhere, because the question makes little sense otherwise. RonL
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