Final answer:
To prove Chebyshev's theorem, (X-μ)² is substituted for X into the provided inequality, allowing the assessment of the probability that a random variable falls within a particular range of the mean in terms of standard deviations.
Step-by-step explanation:
To prove Chebyshev's theorem using the inequality of Exercise 29, we need to substitute (X-μ)² for X. Chebyshev's theorem relates to the probability that a random variable falls within a certain number of standard deviations of the mean. By substituting (X-μ)² into the inequality provided, one applies Chebyshev's theorem to the distribution of X², which in this context represents the variance of the random variable X around the mean μ.
In a case where we complete the square in x² and find that 2(x² - 1)² ≤ 4, this can relate back to the standard deviation and the mean of X. Following the application of Chebyshev's theorem, we could then determine the likelihood that a certain observation lies within a given range, often expressed in terms of μ ± kσ (mean plus or minus a multiple of the standard deviation).
Therefore, by making the substitution and leveraging the given inequality, we can apply Chebyshev's theorem to the variance and standard deviation of this distribution of X, demonstrating with examples how it predicts the behavior of the distribution within a certain range of the mean.