COMS 5583 COMPREHENSIVE EXAM STUDY GUIDE WITH COMPLETE SOLUTION

COMS 5583 COMPREHENSIVE EXAM STUDY
GUIDE WITH COMPLETE SOLUTIONS

●● (8.1) M/M/1 queue
Answer: queue length having a single server.


●● (8.3) If the expected value of your estimator equals the parameter
that you're trying to estimate, then your estimator is unbiased. True of
False
Answer: True. This is the definition of unbiasedness


●● (8.3) If X1, X2, ..., Xn are i.i.d. with mean mu, then the sample mean
X-bar is unbiased for mu. True or False
Answer: True.


●● (8.4) What is the MSE (Mean Squared Error) of an estimator?
Answer: Bias^2 + Variance


●● (8.3) What is the expected value of the mean of a Pois(λ) random
variable?
Answer: λ is the mean and the variance

, ●● (8.3) What is the expected sample variance s^2 of a Pois(λ) random
variable?
Answer: λ is the sample variance and the mean


●● (8.4) Suppose that estimator A has bias = 3 and variance = 12, while
estimator B has bias -2 and variance = 14. Which estimator (A or B) has
the lower mean squared error?
Answer: B is lower. Bias^2 + Variance: 18 < 21


●● MLE
Answer: Maximum Likelihood Estimator - "A method of estimating the
parameters of a distribution by maximizing a likelihood function, so that
under the assumed statistical model the observed data is most probable."


●● (8.4) Suppose that X1=4, X2=3, X3=5 are i.i.d. realizations from an
Exp(λ) distribution. What is the MLE of λ?
Answer: 0.25


●● (8.5/8.6) If X1=2, X2=−2, and X3=0 are i.i.d. realizations from a
Nor(μ , σ^2) distribution, what is the value of the maximum likelihood
estimate for the variance σ^2?
Answer: 8/3. MLE of σ^2 is the summation of the squared differences
(Xi - μ), all divided by n.