Let [math]X[/math] be a continuous random variable with mean [math]\mu = 10[/math] and variance [math]\sigma^2 = 100/3[/math]. Using Chebyshev's Inequality, find an upper bound for the following probabilities.

• [math]P(|X - 10| \geq 2)[/math].
• [math]P(|X - 10| \geq 5)[/math].
• [math]P(|X - 10| \geq 9)[/math].
• [math]P(|X - 10| \geq 20)[/math].

Let [math]X[/math] be a continuous random variable with values unformly distributed over the interval [math][0,20][/math].

• Find the mean and variance of [math]X[/math].
• Calculate [math]P(|X - 10| \geq 2)[/math], [math]P(|X - 10| \geq 5)[/math], [math]P(|X - 10| \geq 9)[/math], and [math]P(|X - 10| \geq 20)[/math] exactly. How do your answers compare with those of Exercise? How good is Chebyshev's Inequality in this case?

Let [math]X[/math] be the random variable of Exercise.

• Calculate the function [math]f(x) = P(|X - 10| \geq x)[/math].
• Now graph the function [math]f(x)[/math], and on the same axes, graph the Chebyshev function [math]g(x) = 100/(3x^2)[/math]. Show that [math]f(x) \leq g(x)[/math] for all [math]x \gt 0[/math], but that [math]g(x)[/math] is not a very good approximation for [math]f(x)[/math].

Let [math]X[/math] be a continuous random variable with values exponentially distributed over [math][0,\infty)[/math] with parameter [math]\lambda = 0.1[/math].

• Find the mean and variance of [math]X[/math].
• Using Chebyshev's Inequality, find an upper bound for the following probabilities: [math]P(|X - 10| \geq 2)[/math], [math]P(|X - 10| \geq 5)[/math], [math]P(|X - 10| \geq 9)[/math], and [math]P(|X - 10| \geq 20)[/math].
• Calculate these probabilities exactly, and compare with the bounds in (b).

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?

If [math]X[/math] is normally distributed, with mean [math]\mu[/math] and variance [math]\sigma^2[/math], find an upper bound for the following probabilities, using Chebyshev's Inequality.

• [math]P(|X - \mu| \geq \sigma)[/math].
• [math]P(|X - \mu| \geq 2\sigma)[/math].
• [math]P(|X - \mu| \geq 3\sigma)[/math].
• [math]P(|X - \mu| \geq 4\sigma)[/math].

Now find the exact value using the program NormalArea or the normal table in Appendix A, and compare.

If [math]X[/math] is a random variable with mean [math]\mu \ne 0[/math] and variance [math]\sigma^2[/math], define the relative deviation [math]D[/math] of [math]X[/math] from its mean by

• Show that [math]P(D \geq a) \leq \sigma^2/(\mu^2a^2)[/math].
• If [math]X[/math] is the random variable of Exercise Exercise, find an upper bound for [math]P(D \geq .2)[/math], [math]P(D \geq .5)[/math], [math]P(D \geq .9)[/math], and [math]P(D \geq 2)[/math].

Let [math]X[/math] be a continuous random variable and define the standardized version [math]X^*[/math] of [math]X[/math] by:

• Show that [math]P(|X^*| \geq a) \leq 1/a^2[/math].
• If [math]X[/math] is the random variable of Exercise, find bounds for [math]P(|X^*| \geq 2)[/math], [math]P(|X^*| \geq 5)[/math], and [math]P(|X^*| \geq 9)[/math].

• Suppose a number [math]X[/math] is chosen at random from [math][0,20][/math] with uniform probability. Find a lower bound for the probability that [math]X[/math] lies between 8 and 12, using Chebyshev's Inequality.
• Now suppose 20 real numbers are chosen independently from [math][0,20][/math] with uniform probability. Find a lower bound for the probability that their average lies between 8 and 12.
• Now suppose 100 real numbers are chosen independently from [math][0,20][/math]. Find a lower bound for the probability that their average lies between 8 and 12.

A student's score on a particular calculus final is a random variable with values of [math][0,100][/math], mean 70, and variance 25.

• Find a lower bound for the probability that the student's score will fall between 65 and 75.
• If 100 students take the final, find a lower bound for the probability that the class average will fall between 65 and 75.