Question 8.2.5: Let X be a continuous random variable with values normally d......
Let X be a continuous random variable with values normally distributed over (−∞,+∞) with mean µ = 0 and variance σ2 =1.
(a) Using Chebyshev’s Inequality, find upper bounds for the following probabilities: P(|X|≥ 1), P(|X|≥ 2), and P(|X|≥ 3).
(b) The area under the normal curve between −1 and 1 is .6827, between −2 and 2 is .9545, and between −3 and 3 it is .9973 (see the table in Appendix A). Compare your bounds in (a) with these exact values. How good is Chebyshev’s Inequality in this case?
(a) 1, 1/4, 1/9
(b) 1 vs. .3173, .25 vs. .0455, .11 vs. .0027
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