---
tags: 機率與統計
title: 第三週活動
---
# 第三週-課堂活動

### (一) 機率分布練習
(1) A mail-order computer business has six telephone lines. Let x denote the number of lines at use at a specified time. Suppose the mass function of x is given by:

x 0 1 2 3 4 5 6

p(x) .10 .15 .20 .25 .20 ? ?

(a) In the long run, what proportion of the time will at most 3 lines be in use? Fewer than three lines?
>$0.1+0.15+0.2+0.25=0.7$
>$0.1+0.15+0.2=0.45$

(b) In the long run, what proportion of the time will at least five lines be in use?
>$1-0.1-0.15-0.2-0.25-0.2=0.1$

(c) In the long run, what proportion of the time will between two and four lines, inclusive, be in use?
>$0.2+0.25+0.2=0.65$

(d) In the long run, what proportion of the time will at least four lines not be in use?
>$0.1+0.15+0.2=0.45$
>
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(2) A continuous variable X is said to have a uniform distribution if the density function is given by


The corresponding density "curve" has constant height over the interval from a to b. Suppose the time (min) taken by a clerk to process a certain application form has a uniform distribution with a = 4 and the b = 6.

(a) In the long run, what proportion of forms will take between 4.5 min and the 5.5 min to process? At least 4.5 min to process?
>$(5.5-4.5)*0.5=0.5$
>$(6-4.5)*0.5=0.75$

(b) What value separates the slowest 50% of all processing times from the fastest 50% (the median of the distribution)?
>$(6-4)/2+4=5$

(d) What value separates the best 10% of all processing times from the remaining 90%?
>$(6-4)/10+4=4.2$

### (二) 期望值：平均數與變異數

(1) Let x denote the amount of time for which a book on two-hour reserve at a college library is checked out by a student, and suppose that x has density function f(x)=.5x for 0 < x < 2.

(a) What is the mean value of X? Why is the mean value not 1, the midpoint of the interval of positive density?
>$\int f(x)dx=\int0.5x dx=0.25x$
>$2*0.25=0.5$
>因為不是平均的

(b) What is the median of this distribution, and how does it compare to the mean value?
>中位數 1 平均數 0.5
>中位數>平均數

(c) What proportion of checkout times are within one-half hour of the mean time? What proportion are within one-half hour of the median time？
>$f(0.5)=0.5*0.5=0.25$
>

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(2) Let x represent the number of underinflated tires on an automobile of a certain type, and suppose that p(0)= .4, p(1)=p(2)=p(3)=.1, and p(4)= .3.

(a) Calculate the mean of the distribution.
>$0.4*0+0.1*(1+2+3)+0.3*4=1.8$
(b) Calculate the standard deviation of x.
>$(0.4+0.1*3+0.3)/5=0.2$
>$(0.4-0.2)^2+(0.1-0.2)^2*3+(0.3-0.2)^2=0.08$
>$\sqrt0.08=0.28$

(c) For what proportion of such cars will the number of underinflated tires be within one standard deviation of the mean value? More than three standard deviations from the mean value?
>$0.4+0.1>0.2+0.28$

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第1題: 警方計劃在市區內四個不同位置使用雷達陷阱來取締超速。在L1, L2, L3和L4每個位置的雷達陷阱將有40%，30%，20%和30%的時間在運作。如果一個人在上班途中開快車，分別有0.2， 0.1，0.5和0.2的機率經過上述其中一個位置，

(1) 則他會收到超速罰單的機率為？
(2) 如果一個人在上班途中收到一張超速罰單，則他經過位於L2的雷達陷阱的機率為？

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第2題: 在美國太空計劃的一個支援系統中，一個關鍵零件只有85%的時間正常運作。為了增強系統的可靠度，決定要將三個零件安裝成並聯，使得系統只有當他們全部失效時才會失效。假設零件獨立的運作，且他們三個全部都有85%的成功率，就此點而言他們是等效的。

考慮隨機變數X為三個之中失效的零件數量。

(1) 寫出隨機變數X的機率函數

(2) E(X)=? (也就是說三個之中失效的零件平均數)

(3) Var(X)=?

(4) 整體系統成功運作的機率=?

(5) 系統失效的機率=?

(6) 如果希望系統有0.99的機率成功運作，三個零件足夠嗎? 如果不夠需要多少個?

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第3題. 某種工業公司分配在環境與污染控制上的預算比例即將被稽查。一個資料收集專案求得這些比例的分布為:

f(y)={5(1−y)^4^，0≤y≤1 && 0，其他}

(1) 驗證以上為有效的機率密度函數

(2) 隨機選出的一間公司花在環境與污染控制的經費少於其預算10%的機率=?

(3) 隨機選出的一間公司花在環境污染控制上的經費超過其預算50%的機率=?