# 2009mid ビジネス確率モデル toyo_classes

## 全文

(1)

### 6/4/2009

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.