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Question 6.12: Prove that there does not exist a random variable X with E(X......

Prove that there does not exist a random variable X with E(X) = 𝜇 and Var(X) = 𝜎² such that

P(𝜇 − 2𝜎 < X < 𝜇 + 2𝜎) = 0.70.

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From Chebyshev’s inequality we have, for any variable X with mean 𝜇 and variance 𝜎²,

P(|X-\mu|\geq2\sigma)\leq{\frac{\sigma^{2}}{(2\sigma)^{2}}}={\frac{1}{4}},

which in turn implies that

P(\mu-2\sigma\lt X\lt \mu+2\sigma)=P(|X-\mu|\lt 2\sigma)=1-P(|X-\mu|\geq2\sigma)\geq1-{\frac{1}{4}}={\frac{3}{4}}\gt 0.70.

This shows that the equation in the statement of the example is impossible for any random variable X.

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