This homework is a check to ensure you have the appropriate prereqs for the course. You must write up
your homework using R Markdown. Since you are not doing any computations, you may turn in only the
nal le produced (pdf etc), you don’t need to turn in your code (for this homework). Clearly show all of
your steps and work.
1. Let X be an exponential random variable with rate parameter 3. Then X has probability density
function
f(x) = 13e x=3 for x 0:
(a) Find the mean and variance of X.
(b) Find the 3-rd moment of X: E(X3).
(c) Find P(X2[2;10]).
2. Let X1 exp(1) and X2 exp(1) be independent and identically-distributed exponential random
variables with rate 1. Let:
Y = X1 + X2 ; Z = X1 X2
(a) What is the cdf of X1?
(b) What is the joint pdf of (X1;X2)?
(c) What is the joint pdf of (Y;Z)?
(d) What is the marginal pdf of Z?
3. Clearly state the following:
(a) The central limit theorem (CLT).
(b) The weak law of large numbers (WLLN).
4. Suppose that X1;X2;:::;Xn are iid random variables with pdf:
(a) Find the maximum likelihood estimate of the parameter a.
(b) Find the Fisher Information of X1;X2;:::;Xn and use it to estimate a 95% con dence interval
on the MLE of a.
(c) Explain how the central limit theorem relates to (b).
5. A lab blood test is 95% e ective in detecting a certain disease when the disease is present. However,
if a healthy individual is tested, there is a 1% chance that the test result will imply that she/he has
the disease. If it is known that 5% of the population have the disease, what is the probability that a
person has the disease, given that the test says they do