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math-340:m340-f15-hw:hw-28 [2015/11/12 11:54] dcs created |
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- Let $X$ be binomial with parameters $n = 4$ and $p = \frac{1}{2}$. Use Chebyshev's inequality to find an upper bound $P(|X - 2| \ge 2)$ and compare with the exact probability. | - Let $X$ be binomial with parameters $n = 4$ and $p = \frac{1}{2}$. Use Chebyshev's inequality to find an upper bound $P(|X - 2| \ge 2)$ and compare with the exact probability. | ||

- | - If $Z$ is standard normal, find an upper bound for $P(|Z| \ge 4)$ using Markoff's inequality, Chebyshev's inequality, and Chernoff's inequality. | + | - If $Z$ is standard normal, find an upper bound for $P(|Z| \ge 4)$ using Markov's inequality, Chebyshev's inequality, and Chernoff's inequality. |

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+ | Answers: | ||

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+ | - By Chebyshev's inequality, $P(|X - 2| \ge 2) \le \frac{1}{4}$. Exactly, $P(|X - 2| \ge 2) = \frac{1}{8}$. | ||

+ | - Markov's inequality: $P(|Z| \ge 4) \le \frac{1}{4}\sqrt{\frac{2}{\pi}} = 0.1995$; Chebysev's inequality: $P(|Z| \ge 4) \le \frac{1}{16} = 0.06250$; Chernoff's inequality: $P(|Z| \ge 4) \le 2e^{-8} = 0.006709$ |