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Prelim_Exam

# Prelim_Exam - Stefan Weber Shirshendu Chatterjee ORIE...

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Stefan Weber Shirshendu Chatterjee ORIE, Cornell University Exam 1 ORIE 3500/5500: Engineering Probability and Statistics II Question 1 [25 points] (i) Define the following notions: (a) Independence of the events A and B . (3) (b) CDF of a random variable X . (3) (c) PDF of an absolutely continuous random variable Y . (3) (d) Expected value of a discrete random variable taking finitely many values. (3) (ii) State the following axioms or theorems and explain how these can be used in engineering models: (a) The probability axioms. (3) (b) The total probability theorem. (5) (c) Bayes’ theorem. (5) (Remark: When you answer the questions above, please provide a precise mathematical state- ment as well as a short intuitive explanation.) Question 2 [25 points] (i) Calculate the expectation and the variance of the following distributions: (a) Bernoulli distribution with parameter p (0 , 1). (4) (b) Gaussian distribution with parameters μ R and σ 2 > 0. (6) (ii) Calculate the mean of a Poisson random variable with parameter λ > 0. (5)

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