推 TINTINH:已收精華區:) 03/12 07:53
課程名稱︰統計學
課程性質︰系定必修
課程教師︰林惠玲老師
開課學院：社會科學院
開課系所︰經濟學系
考試日期（年月日）︰2008年1月15日
考試時限（分鐘）：180分鐘
是否需發放獎勵金：是 謝謝^^
（如未明確表示，則不予發放）
試題 :
一、是非題(18分)(請先回答是或非，再說明理由，未說明理由者不予計分)
1.若X為某商品的需求量，Y為其價格，均為隨機變數，則請判斷下列是否正確。
a.若f(X|Y)=f(X)，則X與Y獨立。
b.若Cov(X,Y)=0，則X與Y獨立。
c.若X=a+bY，則ρxy=1。
d.若Y=P0(常數)，則ρxy=0。
e.若X與Y的計量單位改變，則Cov(X,Y)均不會改變。
2.設X為隨機變數，其平均數為μ，變異數為σ^2，今由該母體以簡單隨機抽樣，抽取
(X1,......,Xn)為樣本，請判斷下列敘述是否為正確。
_ _ _
a.令X=ΣX/n，X在估計μ時，X為所有估計式中變異數最小的。
_
b.Sx^2=Σ(X-X)^2/n-1，則Sx^2為σ^2的不偏誤估計式。
c.Sx^2為σ^2的一致性估計式。
d.Sx為σ的不偏且為一致性估計式。
二、問答題(16分)
(1)請以統計觀念解釋：100家廠商專利權數的總合，接近常態分配；但廠商的
的專利權數經常為一Possion分配。(2分)
(2)設X與Y為二元隨機變數，W=aX+bY，S=cX+dY，試求Cov(W,S)。(4分)
(3)設X1,X2是從N(μ,σ^2)隨機抽樣的兩樣本，則(X1-X2)^2/2σ^2的分配為何，
請證明之。(3分)
(4)欲知96年6月畢業生的失業率，若想自行調查研究，請問應如何抽樣？(請回答抽樣
底冊為何？應採簡單隨機抽樣，還是分層抽樣？請簡單說明其理由以及抽樣的步驟
。)(4分)
(5)承上題，在95%信賴水準下，若想使抽樣誤差在2%以內，請問他至少要抽樣多少個？
(3分)
(未學過微積分的同學，請回答第四題，否則答第三題，答了第三題不必回答第四題)
三、(15分)Let X and Y have a joint density function given by
f(x,y)=3x 0 ≦y＜x≦1
=0 otherwise
a. Find the marginal density function of Y.(3分)
b. Find f(Y|X=x).(3分)
c. Find E(Y|X=x), V(Y|X=x).(6分)
d. Explain what's the difference between E(Y|X=x) and E(Y)，V(Y|X=x) and
V(Y).(3分)
四、(15分)Let X and Y have a joint density function given by
f(x,y)=1/21(x^2+y) x=1,2; y=0,1,2
a. Find the marginal probablity function of Y.(3分)
b. Find f(Y|X=x).(3分)
c. Find E(Y|X=x), V(Y|X=x).(6分)
d. Explain what's the difference between E(Y|X=x) and E(Y)，V(Y|X=x) and
V(Y).(3分)
五、(6分)Each customer who enters Larry's clothing store will purchase a suit
with probablity of 0.2. Assume that the number of customers entering the
store in one day is Poisson distributed wiyh mean of 50. Let X denote the
number of suits Larry sold and Y denote the number of customers entered
Larry's store.
Use E(X)=E(E(X|Y)) and Var(X)=Var(E(X|Y))+E(Var(X|Y)) to find E(X) and
Var(X).
六、(10分)The amount of fill dispensed by a bottling machine is normally
distributed with σ=1 ounce. If n=9 bottles are randomly selected from
the output of the machine,
a. What the probablity that he sample mean will be within 0.3 ounce of the
true mean?
_
b. When n is larger, what will happen to the value of P(│Y-μ│≦0.3)？
What's the implication of the result?(3分)
c. Find P(S^2≧2).(3分)
七、(15分)劉華發現他打電話訂位飛機票時，經常是忙線中，他想估計打電話訂位
"成功"的機率為何，於是，他蒐集10次訂位的記錄，得到訂位成功打電話的次數
如下：
1, 3, 2, 4, 3, 4, 3, 2, 2, 1
(hint:該母體為幾何分配，其機率函數為
f(x)=P(1-P)^x-1 x=1,2,...
=0 其他
其中P為打電話訂位成功的機率，X為打電話直到訂位成功的次數)
試回答下列小題:
(下面(1)、(2)小題任選一題回答，其他(3)、(4)小題必須回答)
(1)請利用最大概似法求打電話訂位成功的機率(P)的最大概似估計式(MLE)及估計值
。(6分)
(2)請利用動差法求P的動差估計式(MME)及估計值。(6分)
(3)試求P(X≧3)的估計值。(3分)
(4)試問你所得到的P的估計式是一不偏與一致性估計式嗎？理由。(6分)
八、(20分)A survey found that households tend to spend an average of
NT$17,576 for food and beverage during Chinese Lunar New Year in a small
town. Assume that the survey included 25 households and the standard
deviation was NT$4,470.
a. With 95% confidence interval, what is the estimation of error of the
population mean?(What assumptions are necessary to ensure the validity
of the estimation?)
b. What is the 95% confidence interval estimate of the population mean?
c. What is the 95% confidence interval estimate of the population standard
deviation?
d. The prior year, the population mean expenditure per household was NT
$17,116. Discuss the change in Chinese Lunar New Year expenditure over
one-year period.
e. Discuss skewness that may be present in the population. What suggestion
would you make for a repeat of this study?
f. If the researcher want to have 95% confidence of estimating the
population mean expenditure within $1,000 and the sandard deviation is
assumed to be $4,000, what sample size is needed?
g. If there is one thousand households in the small town, what is the
confidence interval of total expenditure?(C.C.=95%)
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