Exercises 10.7: 1, 2

Hints: For Exercise 1, note that $f(p) = p(1 - p)$ is maximized on $[0, 1]$ when $p = \frac{1}{2}$.

For Exercise 2, the sample mean of $X_1, X_2, \ldots, X_n$ is $\bar{X} = \dfrac{X_1 + X_2 + \cdots + X_n}{n}$.

Also:

1. 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.
2. 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.

Answers:

1. By Chebyshev's inequality, $P(|X - 2| \ge 2) \le \frac{1}{4}$. Exactly, $P(|X - 2| \ge 2) = \frac{1}{8}$.
2. 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: $P(|Z| \ge 4) \le 2e^{-8} = 0.006709$