Chebyshev's inequality (Sect. 3.2) states that for any...

Chebyshev's inequality (Sect. 3.2$)$ states that for any number $k$ satisfying $k \geq 1, P(|X-\mu| \geq k \sigma)$ is no more than 1$/ k^{2} .$ Obtain this probability in the case of a normal distribution for $k=1,2,$ and $3,$ and compare to Chebyshev's upper bound.

Key Concepts

Chebyshev's Inequality

Chebyshev's Inequality is a fundamental result in probability theory that provides an upper bound on the probability that a random variable deviates from its mean by a specified number of standard deviations. Regardless of the underlying distribution (provided the variance exists), it guarantees that the probability of a deviation of k standard deviations or more is at most 1/k². This result is particularly useful when little is known about the actual distribution since it is universally applicable.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, with its shape characterized by the bell curve. It is defined by its mean and standard deviation, and many natural and measurement phenomena are approximately normally distributed due to the central limit theorem. This distribution has well-known properties and precise formulas for probabilities that make it a benchmark in statistics.

Tail Probabilities

Tail probabilities refer to the likelihood that a random variable falls in the far ends ('tails') of the distribution, far from the mean. In many contexts, particularly in risk assessment and statistical hypothesis testing, assessing these probabilities is crucial because they quantify the chances of extreme deviations from the average behavior.

Empirical Rule

The Empirical Rule, often associated with the normal distribution, states that approximately 68% of the data lie within one standard deviation of the mean, about 95% within two standard deviations, and roughly 99.7% within three standard deviations. This rule offers a quick, intuitive way to understand the distribution of data and the probabilities of tail events in normal distributions.

Comparison of Theoretical Bounds and Exact Probabilities

Comparing theoretical bounds like Chebyshev's inequality with exact probabilities computed for specific distributions, such as the normal distribution, highlights the conservative nature of general inequalities. While Chebyshev's inequality applies broadly, it often overestimates the probability of extreme deviations when additional information (such as normality) is available, as the actual tail probabilities in a normal distribution are significantly lower than the bound provided by Chebyshev's inequality.

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