3. Applying Chebyshev's Inequality. Chebyshev's inequality states that no more than \( \frac{1}{k^2} \) of the distribution's values can be more than \( k \) standard deviations away from the mean, for any \( k > 1 \). In this exercise, you'll use Chebyshev's inequality to estimate the probability that a random variable deviates from its mean by more than 5 standard deviations.

You are given the formula \( P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2} \), where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

If the standard deviation of an asset's return is 10\%, what is the maximum probability of observing a return that is 20\% above or below the mean?

# Calculate distance from mean in standard deviations. k <- 20 / ___ # Use Chebyshev's inequality to calculate the upper bound. probability_upper_bound <- ___/k^___ # Print probability upper bound. probability_upper_bound # Calculate distance from mean in standard deviations. k <- 20 / 10 # Use Chebyshev's inequality to calculate the upper bound. probability_upper_bound <- 1/k^2 # Print probability upper bound. probability_upper_bound test_error() test_object("probability_upper_bound", incorrect_msg="Ensure you've used the formula correctly with k = 20 / 10.") success_msg("Excellent work! The probability of observing a value more than 2 standard deviations from the mean is very low.")