1.

Chebyshev’s inequality gives . An exact calculation yields instead . To see this, note that and so that

2.

3.

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 .

4.

a)

Applying our findings from Question 3,

b)

TODO (Computer Experiment)

c)

The length of the interval is . This length is at most if and only if .

TODO (Plot)

5.

As per the hint,

6.

TODO (Plot)

7.

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 .