【統計學】第五章

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思考題

1.頻率與概率有什么關系？

事件A的概率是描述事件A在試驗中出現的可能性的大小的一種度量，記事件A出現的可能性大小的數值為P(A),P(A)稱為事件A的概率。
頻率更接近實驗值，概率更接近理論值。

2.獨立性與互斥性有什么關系？

獨立性：兩個事件不論哪一個事件發生並不影響另一個事件發生的概率，則稱這兩個事件相互獨立。
互斥性：互斥事件一定是相互依賴(不獨立)的，但是相互依賴的事件不一定是互斥的。
不互斥事件可能是獨立的，也可能是不獨立的，然而獨立事件不可能是互斥的

3.根據自己的經驗體會舉幾個服從泊松分布的隨機變量的實例

1.在某企業中每月發生的事故的次數
2.單位時間內到達某一服務櫃台(服務站、診所、超市的收銀台、電話總機等)需要服務的顧客人數
3.人壽保險公司每天收到的死亡聲明的個數
4.某種儀器每月出現故障的次數

4.根據自己的經驗體會舉幾個服從正態分布的隨機變量的實例

1.某地區同年齡組兒童的發育特征，如身高、體重、肺活量
2.某公司年銷售量
3.在同一條件下產品的質量以平均質量為中心上下擺動

練習題

寫出下列隨機試驗的樣本空間

1.記錄某班一次統計學測驗的平均分數
某班人數
2.某人在公路上騎自行車，觀察該騎車人在遇到第一個紅燈停下來以前遇到的綠燈次數
所有燈
3.生產產品，直到有10件正品為止，記錄生產產品的總件數
產品總數

某市有50%的住戶訂日報，有65%的住戶訂晚報，有85%的住戶至少訂兩種報紙中的一種，求同時訂這兩種報紙的住戶的百分比。

0.5+0.65−0.85=0.3

設A與B是兩個隨機事件，已知A與B至少有一個發生的概率是1/3，A發生且B不發生的概率是1/9，求B發生的概率

P(B)=31​−91​=92​

有甲、乙兩批種子，發芽率分別是0.8和0.7。在兩批種子中隨機抽取一粒，求1.兩粒都發芽的概率；2.至少有一粒發芽的概率；3.恰有一粒發芽的概率

1.0.56
2.0.94
3.0.38

某廠產品的合格率為96%，合格品中一級品率為75%，從產品中任取一件為一級品的概率為？

P(AB)=P(A)⋅P(A∣B)=0.96∗0.75=0.72

某種品牌的電視機用到5000小時未壞的概率為3/4，用到10000小時未壞的概率為1/2.現在有一台這種品牌的電視已經用了5000小時未壞，它能用到10000小時的概率？

P(A)=21​P(AB)=43​P(B)=32​

考慮擲兩枚硬幣的試驗。令X表示觀察到正面的個數，試求X的概率分布。

X	結果	概率
2	正正	0.25
1	正反	0.25
1	反正	0.25
0	反反	0.25

某人花2元錢買彩票，他抽中100元獎的概率是0.1%，抽中10元獎的概率是1%，抽中1元獎的概率是20%，假設各種獎不能同時抽中，試求

1.此人收益的概率分布

2.此人收益的期望值

結果	概率
抽中100元	0.1%
抽中10元	1%
抽中1元	20%

期望值：0.4

設隨機變量X的概率密度為：

f(x)=θ33x2​,0<x<θ

(1)已知$P(X>1)=\frac{7}{8}%,求 θ的值
(2) 求X的期望值和方差

87​=∫1θ​θ33x2​dx

θ=2

E(x)=∫02​x83x2​dx=23​

E(x)=∫02​x283x2​dx=512​

D(x)=Ex2−(Ex)2=203​

一張考卷上有5道題目，同時每道題列出4個備選答案，其中有一個答案是正確的。某學生憑猜測能答對至少4道題的概率是多少？

41​5+5∗43​∗41​4=641​

設隨機變量X服從參數為λ的泊松分布，且已知P{X=1}=P{X=2},求P{X=4}

泊松分布公式

P(X=k)=k!λk​e−λ

λ=2帶入公式

PX=4=32​e−2

一工廠生產的電子管壽命X(以小時計算)服從期望值μ=160的正態分布，若要求P{120<X<200}>=0.08,允許標准差最大為多少？

工廠生產的電子管壽命在120＜X＜200之間時的概率為 P{120＜X＜200} ＝ P{(120－160)／δ＜ (X－120)／δ＜ (200－160)／δ} ＝ P{-40／δ＜ Z ＜40／δ} ＝φ(40／δ)－φ(-40／δ) ＝ 2φ(40／δ)－1 ≥ 0.08 →φ(40／δ) ≥0.54 通過查閱標准正態分布表得 40／δ≥0.11 δ≤363.64

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