Question

Let X be a random variable representing the length of an episode of Andor, in minutes. Initially, we don't know anything about the PDF of X, other than that it is symmetric about its mean.

(a) Suppose we learn that E(X) = 50. Compute the Markov bound on P(X ≥ 55).

(b) We now further learn that Var(X) = 16. Compute the Chebyshev bound on P(|X - 50| ≥ 5), and then divide this by two to get an estimate of P(X ≥ 55) since we know the distribution is symmetric.

(c) Finally, we learn the complete distribution of X: it's actually a Gaussian with the mean and variance given above. Compute the actual value of P(X ≥ 55). How does it compare to the bounds above?

Answer 1

Calculate the Markov and Chebyshev bounds for the probability of the length of an episode of Andor being at least 55 minutes, based on the given mean and variance. For an exact probability, use the Gaussian distribution with these parameters.

Step-by-step explanation:

The question requires using knowledge of probability and statistics to calculate the bounds and actual probabilities regarding the length of an episode of Andor. Given that X represents the length of an episode, when we know the mean (E(X)) is 50, we can determine the Markov bound on P(X ≥ 55). However, knowing further that the variance (Var(X)) is 16, we can use the Chebyshev inequality to find the bound on P(|X - 50| ≥ 5) and then adjust for the probability of P(X ≥ 55) based on the symmetric distribution assumption. When the complete distribution is identified as a Gaussian (or normal) distribution with the given mean and variance, we can calculate the exact probability of P(X ≥ 55) using the standard normal distribution table or a calculator.