STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 5 8 Markov s Inequality Let u X be a non negative function of the random variable X If E u X exists then for every positive constant c P u X c E u X c Chebyshev s Inequality Let X be any random variable with mean and variance 2 For any 0 2 P X 2 or equivalently 2 P X 1 2 Setting k k 1 we obtain P X k 1 k2 or equivalently P X k 1 1 k2 That is for any k 1 the probability that the value of any random variable will 1 be within k standard deviations of its mean is at least 1 k2 Example 1 Suppose E X 17 SD X 5 Consider interval 9 25 17 8 17 8 P 9 X 25 P X 1 6 1 1 1 6 2 k 8 1 6 5 39 0 609375 64 Suppose E X 17 SD X 5 Example 2 Suppose also that the distribution of X is symmetric about the mean Consider interval 10 30 17 7 17 13 1 4 2 6 P 10 X 24 P X 1 4 1 P 4 X 30 P X 2 6 1 1 1 4 2 1 2 6 2 0 490 0 852 Since the distribution of X is symmetric about the mean 0 490 0 852 P 10 X 17 0 245 P 17 X 30 0 426 2 2 P 10 X 30 0 245 0 426 0 671 Example 3 Consider a discrete random variable X with p m f P X 1 E X 0 Then P X 1 2 Var X E X 2 1 P X P X 1 1 k 1 P X P X 1 0 Chebyshev s Inequality cannot be improved Example 4 Let a 0 0 p P X a p E X 0 Then Let k 1 2p Consider a discrete random variable X with p m f P X 0 1 2 p P X a p 2 Var X E X 2 2 p a 2 1 Then k a P X k P X a 2 p 1 k2 P X k P X a 1 2 p 1 1 k 2
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