Chebyshev’s inequality gives . An exact calculation yields instead . To see this, note that and so that
First, note that . Chebyshev’s inequality yields
Next, note that
Let so that . Then, and . Hoeffding’s inequality yields
Similarly, . It follows that
is tighter than the Chebyshev bound for sufficiently large .
Applying our findings from Question 3,
TODO (Computer Experiment)
The length of the interval is . This length is at most if and only if .
As per the hint,
A linear combination of IID normal random variables is itself a normal random variable. Therefore, is a random variable with zero mean and variance . Letting , Mill’s inequality yields
The above is tighter than the Chebyshev bound for sufficiently large .