For each of the distributions listed below, use \(\mathrm{R}\) to compute \(P(|X-\mu|>\delta)\) and compare the result to the bound given by Theorem 2.7 as \(\delta^{-2} \sigma^{2}\) for \(\delta=\frac{1}{2}, 1, \frac{3}{2}, 2\). Which distributions become closest to achieving the bound? What are the properties of these distributions?

a. \(\mathrm{N}(0,1)\)

b. \(\mathrm{T}(3)\)

c. \(\operatorname{Gamma}(1,1)\)

d. \(\operatorname{Uniform}(0,1)\)

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Theorem 2.7 (Tchebysheff). Consider a random variable X with E(X) = and V (X) = o < . Then P(|X - | > d) d g.