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BBy Bot
Jun 09'24
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A fair coin is tossed 100 times. The expected number of

heads is 50, and the standard deviation for the number of heads is [math](100 \cdot 1/2 \cdot 1/2)^{1/2} = 5[/math]. What does Chebyshev's Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more standard deviations (i.e., by at least 15)?

BBy Bot
Jun 09'24
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Write a program that uses the function

[math]\mbox {binomial}(n,p,x)[/math] to compute the exact probability that you estimated in Exercise Exercise. Compare the two results.

BBy Bot
Jun 09'24
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Write a program to toss a coin 10,00 times. Let [math]S_n[/math] be the

number of heads in the first [math]n[/math] tosses. Have your program print out, after every 1000 tosses, [math]S_n - n/2[/math]. On the basis of this simulation, is it correct to say that you can expect heads about half of the time when you toss a coin a large number of times?

BBy Bot
Jun 09'24
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A 1-dollar bet on craps has an expected winning of [math]-.0141[/math]. What

does the Law of Large Numbers say about your winnings if you make a large number of 1-dollar bets at the craps table? Does it assure you that your losses will be small? Does it assure you that if [math]n[/math] is very large you will lose?

BBy Bot
Jun 09'24
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Let [math]X[/math] be a random variable with [math]E(X) =0[/math] and [math]V(X) = 1[/math]. What

integer value [math]k[/math] will assure us that [math]P(|X| \geq k) \leq .01[/math]?

BBy Bot
Jun 09'24
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Let [math]S_n[/math] be the number of successes in [math]n[/math] Bernoulli

trials with probability [math]p[/math] for success on each trial. Show, using Chebyshev's Inequality, that for any [math]\epsilon \gt 0[/math]

[[math]] P\left( \left| \frac {S_n}n - p \right| \geq \epsilon \right) \leq \frac {p(1 - p)}{n\epsilon^2}\ . [[/math]]

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Find the maximum possible value for [math]p(1 - p)[/math] if [math]0 \lt p \lt 1[/math]. Using this result and Exercise Exercise, show that the estimate

[[math]] P\left( \left| \frac {S_n}n - p \right| \geq \epsilon \right) \leq \frac 1{4n\epsilon^2} [[/math]]

is valid for any [math]p[/math].

BBy Bot
Jun 09'24
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A fair coin is tossed a large number of times. Does the Law of Large

Numbers assure us that, if [math]n[/math] is large enough, with [math]\mbox {probability} \gt .99[/math] the number of heads that turn up will not deviate from [math]n/2[/math] by more than 100?

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

In Exercise \ref{sec 6.2}., you showed that, for the

hat check problem, the number [math]S_n[/math] of people who get their own hats back has [math]E(S_n) = V(S_n) = 1[/math]. Using Chebyshev's Inequality, show that [math]P(S_n \geq 11) \leq .01[/math] for any [math]n \geq 11[/math].

BBy Bot
Jun 09'24
[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] by any random variable which takes on values 0, 1, 2,

\dots, [math]n[/math] and has [math]E(X) = V(X) = 1[/math]. Show that, for any positive integer [math]k[/math],

[[math]] P(X \geq k + 1) \leq \frac 1{k^2}\ . [[/math]]