[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] be a continuous random variable with values normally

distributed over [math](-\infty,+\infty)[/math] with mean [math]\mu = 0[/math] and variance [math]\sigma^2 = 1[/math].

• Using Chebyshev's Inequality, find upper bounds for the following probabilities: [math]P(|X| \geq 1)[/math], [math]P(|X| \geq 2)[/math], and [math]P(|X| \geq 3)[/math].
• The area under the normal curve between [math]-1[/math] and 1 is .6827, between [math]-2[/math] and 2 is .9545, and between [math]-3[/math] 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?