2009mid OR toyo_classes

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Operations Research

Midterm

2009

Hiroshi Toyoizumi

6/1/2009

1. What is OR? Explain it, using examples.

2. Suppose you are tested by a disease that strikes 1/1000 population. This test has 5% false positives, that mean even if you are not affected by this disease, you have 5% chance to be diagnosed to be suffered by it. A medical operation will cure the disease, but of course there is a mis-operation. Given that your result is positive, what can you say about your situation?

3. Let X be a Bernouilli random variable with P[X = 1] =p and P[X = 0] = 1−p. Then find

E[X] =, (1)

V ar[X] =, (2)

using the fact X2 =X for Bernouille random variables.

4. Describe the relationship of Markov chain and Google.

5. What is the birth and death process? What kind of features do Poisson processes have?

6. Let X(t) be a Poisson process. Given that

Pk(t) =P{X(t) =k}= (λt)k

k! e

−λt, (3)

find E[X(t)].

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7. You can write anything you want.

Remark 1. Don’t write lengthy answers. Your answers should be concise

and focused.

Remark 2. Each problem is 5 point worth.

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