---
tags: 機率與統計
title: 第五週活動
---
# 第五週-課堂活動

### 活動一 普瓦松程序，伽瑪與指數分布應用
(1) 假設某車廠維修中心的預約服務來電是依照一個普瓦松程序(Poisson Process)，且平均每分鐘收到2.7通預約電話。求以下事件的機率:
>$p(x,\lambda t)=\frac{e^{-\lambda t}\times (\lambda t)^x}{x!}$

(a) 在一分鐘內不超過4通來電的機率。
>$\sum\limits_{x=0}^4 \frac{e^{-2.7}\times 2.7^x}{x!}=0.8627$

(b) 在5分鐘內超過10通來電的機率。
>$\sum\limits_{x=0}^{10} \frac{e^{-13.5}\times 13.5^x}{x!}=0.7888$

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(2) Data collected at Toronto Pearson International Airport suggests that an exponential distribution with λ=.37 is a good model for rainfall duration in hours. Answer the following questions:

(a) What proportion of rainfall duration at this location are at least 2 hours?
>$\int_{0}^{a}\lambda e^{-\lambda x}dx$
>$\int_{0}^{2}\lambda e^{-\lambda x}dx=0.5228$

(b) What must the duration of a rainfall be to place it among the longest 5% of all times?
>$\int_{0}^{a}\lambda e^{-\lambda x}dx=0.95$
>$-e^{-\lambda a}-(-e^{\lambda 0})=0.95$
>$-e^{-0.37a}+1=0.95$
>$e^{-0.37a}=0.05$
>$-0.37a=ln(0.05)$
>$a=8.09$

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(3) 假設一家銀行只有一個櫃台，提供9個等待區座位。假設服務每個客戶固定需要2分鐘，客戶的平均到達率是5位/分鐘，請問一個客戶進門遇到客滿的機率是多少?
>$\int xe^{-ax}dx=xe^{-ax}-e^{-ax}+c$
>

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### 活動二: 聯合質量函式的應用

A large insurance agency provides service to a number of customers who have purchased the both a homeowner's policy and an automobile policy. For each type of policy, a deductible amount must be specified. Let x denote the homeowner's deductible amount and y denote the automobile deductible amount for a customer who has both types of policies. The joint mass function of a and y is as follows.

joint mass function of f(x, y)
y
f(x, y) 0 250 500
x 200 .20 .10 .20

500 .05 .15 .30

(a) What proportion of customers have $500 deductible amounts for both types of policies ?
>$0.2+0.3=0.5$

(b) What is the marginal mass function of x? What is the marginal mass function of y ?
>$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)dxdy$