14.12.1!1!Stat 100A-Sanchez/Intro probability! 1!Bounds for Probability for one random variable. "Chebyshev’s theorem and Markov’s theorem. Law of large numbers. !Stat 100A-Sanchez/Intro probability! 2!• Introduction to probability bounds. Markov inequality. !• Tchebysheff’s inequality and its applications. !• Law of Large Numbers. Ross, Chapter 8, section 8.1, 8.2, a little of 8.4 Stat 100A-Sanchez/Intro probability! 3!• Up to now, probabilities were arrived at by assuming that the random variable followed a particular distribution. !• What if the distribution of the random variable is unknown? In this lecture, we introduce tools for estimating probabilities when the information on the random variable is limited.!• These new tools consist of assigning bounds to the probabilities. !INTRODUCTION Stat 100A-Sanchez/Intro probability! 4!Markov Inequality: Markov inequality shows how to obtain probability bounds when the mean, but nothing else, is known and X ≥0. If a is a constant, a > 0, "Gives the tail-bound of a nonnegative random variable, when all we know is its expectation or mean. More accurate if the variance is large. "It says that for any value a>0, no matter how large, the probability that the random variable is greater than or equal to a is less than or equal than the expected value divided by a. "€ P(X ≥ a) ≤E (X )a • Markov’s inequality gives the best bound possible for a nonnegative random variable when all that is known is the mean.!• In the following example I compare the tail probability obtained with Markov’s inequality with the tail probability that would be obtained if the distribution of the random variable were known. Stat 100A-Sanchez/Intro probability! 5!Example: The average number of system crashes in a week is 2: E(X) =2. !• Estimate the probability that there will be three or more error crashes in a week in any given week. Compare what you obtain with what you would obtain if you assumed that the variable is Poisson. Stat 100A-Sanchez/Intro probability! 6!€ Markovµ= 2, a = 3 P(X ≥ 3) ≤ 2/3Poisson (Exact probability)P(X ≥ 3) = 1- P(X ≤ 2) = 0.32314.12.1!2!Exercise for you An online computer system is proposed. The !manufacturer can give the information that the mean response time is 10 seconds !E(T) =10 s!(a) Estimate the probability that the response time will be more than 20s.!(b)What would the probability be if it is known that the response time is distributed as usual for this type of random variable? Stat 100A-Sanchez/Intro probability! 7!Stat 100A-Sanchez/Intro probability! 8!• At least 3/4 of the measurements will fall within 2σ of the mean μ, i.e. within!μ± 2σ!• At least 8/9 of the measurements will fall within 3σ of the mean μ, i.e. within! μ± 3σ!If you know sigma and mu, regardless of the shape of the distribution:!Chebychev’s theorem Stat 100A-Sanchez/Intro probability! 9!€ P{ X −µ< kσ} =Pµ− kσ< X <µ+ kσ( )≥ 1−1k2It is from this expression that the guidelines given earlier follow:!If k = 2 1-1/4 = 3/4 !If k = 3 1-1/9 = 8/9 !More generally, Stat 100A-Sanchez/Intro probability! 10!Double check now what we said earlier:!• At least 3/4 of the measurements will fall within 2σ of the mean μ, i.e. within!μ± 2σ!P(μ-2σ< X< μ+2σ)≥3/4!• At least 8/9 of the measurements will fall within 3σ of the mean μ, i.e. within!μ ± 3σ!Regardless of the shape of the distribution:!Stat 100A-Sanchez/Intro probability! 11!Chebyshev’s Theorem !€ P{ X -µ≥ k} ≤σ2k2If X is any finite r.v. with mean μ and variance σ2 "Then for any value k > 0!Stat 100A-Sanchez/Intro probability! 12!Proof. Uses Markov inequality (first line below)!€ P(X ≥ a) ≤E (X )aP{(X −µ)2≥ K2} ≤E (X −µ)2K2but (X −µ)2≥ K2 if X −µ≥ KP( X −µ≥ k) ≤σ2k214.12.1!3!Stat 100A-Sanchez/Intro probability! 13!€ P(X ≥ a) ≤E (X )a Markov's inequalityP{(X −µ)2≥ k2σ2} ≤E (X −µ)2k2σ2P{ X −µ≥ kσ} ≤σ2k2σ2=1k2Generalization of Chebyshev’s theorem:!Stat 100A-Sanchez/Intro probability! 14!Using the complement rule,"€ (1) P{ X -µ≥ k} ≤σ2k2 ⇒ P{ X -µ< k} ≥ 1−σ2k2(2) P{ X -µ≥σk} ≤1k2 ⇒ P{ X -µ< kσ} ≥ 1−1k2Stat 100A-Sanchez/Intro probability! 15!Example: Electric Motor Production!a) What can be said about the fraction of days on which the production level falls between 100 and 140?!b) Find the shortest interval about the mean certain to contain at least 90% of the daily production levels.!The daily production of electric motors at a certain factory averaged 120, with a standard deviation of 10.!Stat 100A-Sanchez/Intro probability! 16!Solution a: Electric Motor Production!The interval from 100 to 140 represents µ-2σ to µ+2σ with µ = 120 and σ = 10.!Thus,! k = 2 and 1-1/k2 = 1-1/4 = 3/4!At least 75% of the days, therefore, will have a total production value that falls in this interval. !Stat 100A-Sanchez/Intro probability! 17!Solution b: Electric Motor Production!To find k, we must set 1-1/k2 = 0.9 and solve for k. !€ 1−1k2= 0.9 1k2= 0.1 k2= 10k = 10 = 3.16The interval µ - 3.16σ to µ + 3.16σ!or 120 - 3.16(10) to 120 + 3.16(10)!!!! 88.4 to 151.6!will contain at least 90% of the daily production levels.!Stat 100A-Sanchez/Intro probability! 18!Example : Computer Breakdowns"The number of computer breakdowns by a university computer system is closely monitored by the director of the computer center, since it is critical to the efficient operation of the center. The number average 4 per week, with a standard deviation of 0.8 per week."a) Find an interval that includes at least 90 percent of the weekly figures for the number of breakdowns."b) The center director promises that the number of breakdown will rarely exceed 8 in a one-week period. Is the director safe in making this claim?!14.12.1!4!Stat 100A-Sanchez/Intro probability! 19!Solution: "a) X = # of weekly breakdowns!€ µx= 4 σx= 0.8P(4 − 0.8k < X < 4 + 0.8k) ≥ 1−1k2= 0.91−1k2= 0.9 k = 10 So, the desired interval is: !€ 4 − 0.8 10 , 4 − 0.8 10( )= 1.4702 , 6.5298( )Stat 100A-Sanchez/Intro probability! 20!Solution:"b) Eight breakdowns is"from the mean. The interval (μ - 5σ, μ + 5σ)"or (0, 8) must contain at least 0.96 of the probability.!Thus, at most 4% of the probability mass can exceed 8 breakdowns and
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