MA/STAT416: ProbabilityLecture NotesSpring 20111Chapter 4: Discrete Random Variables and Mass FunctionsMarch 7, 20114.5Uses ofμandσas Summaries; Markov and Chebyshev’s InequalityTheorem 1(Chebyshev’s Inequality).SupposeE(X) =μandV ar(X) =σ2are as-sumed to be finite. Letkbe any positive number. Then,P(|X-μ| ≥kσ)≤1k2.Example2.Find the variance and standard deviation of the sum of the scores in rollinga fair die twice.•How much of the probability distribution falls within two sigma from the mean?Three sigma from the mean?•Compute the Chebyshev bound for the two sigma and the three sigma intervalaround the mean. Compare the bound with the exact value of those probabilities.Example3.Find the variance and standard deviation of the number of aces in a handof 4 cards drawn randomly from a deck of 52 cards.•How much of the probability distribution falls within two sigma from the mean?
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