Let there be i.i.d. r.v.s. with mean and variance .
Give a value of (a specific number) that will ensure that there is at least a 99% chance that the sample mean will be within 2 standard deviations of the true mean .
Let be the sample mean, which is equal to . Saying that there is at least a 99% chance that is the same as saying that there is at most a 1% chance for . Thus, we want to calculate an such that:
Applying Chebyshev's inequality, we get the following:
Therefore, if we choose , we get the desired inequality.
Explain why a Gamma random variable with parameters is approximately Normal when n is large.
Let . Then, we can have and 's be i.i.d. . By the central limit theorem, since is the sum of i.i.d. random variables, it converges to a normal distribution as .
Let . Determine and such that as
Using the parameters of the Gamma distribution, the central limit theorem also tells us that
In order to convert this normal distribution to a standard normal (), all we need to do is subtract the mean and divide by the standard deviation. Thus:
Suppose is a two-state Markov chain with transition matrix
The rows and columns are indexed 0,1 such that .
Find the stationary distribution of by solving .
By solving , we have that
And by solving this system of linear equations, it follows that
Show that this Markov Chain is reversible under the stationary distribution found in the previous question.
To verify the validity of a stationary distribution for a chain, we just need to show that , which is done if we can show that . We have that
which satisfies our reversibility condition and verifies our stationary distribution from the previous question.
Let . Is a Markov chain? If so, what are the states and transition matrix?
Yes, is a Markov Chain because conditional on , is independent to . This is because the components of and are either constants conditioned in or independent of each other given that is a Markov Chain.
The states are given as . The transition matrix is given as