2009mid ビジネス確率モデル toyo_classes








Performance Evaluation

Midterm 2009

Hiroshi Toyoizumi


1. Using examples, explain the concept of probability space.

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] = pand P[X = 0] = 1−p. Find out

E[X] =?, (1)

V ar[X] =?, (2) using the factX2=X for Bernouille random variables.

4. Describe the relationship of Markov chain and Google.

5. Explain the concept of infinitesimal generator Qof birth and death pro-cesses. What kind of features does the infinitesimal generatorQof Poisson process have?

6. LetX(t) be a Poisson process. Given that

Pk(t) =P{X(t) =k}= (λt)k

k! e

−λt, (3)

find E[X(t)] and V ar[X(t)], using the definition of the expectation and variance.

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 10 point worth.




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