The bernoulli distribution
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The Bernoulli Distribution. 定義. 若一隨機試驗只有兩種課能的結果(正面反面、成功失敗),則此試驗稱之為伯弩利試驗。 A random variable X has a Bernoulli distribution with parameter p (0  p  1) if X can take only the values 0 and 1 and the probabilities are P(X=1) = p P(X=0) = (1-p)

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The Bernoulli Distribution

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The bernoulli distribution

The Bernoulli Distribution

定義

  • 若一隨機試驗只有兩種課能的結果(正面反面、成功失敗),則此試驗稱之為伯弩利試驗。

  • A random variable X has a Bernoulli distribution with parameter p (0  p  1) if X can take only the values 0 and 1 and the probabilities are

  • P(X=1) = p

  • P(X=0) = (1-p)

  • If we let q = 1- p, then the p.f of X can be written as follows:

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The bernoulli distribution1

The Bernoulli Distribution

定義

  • E(X) = 1·p +0·q = p

  • E(X2) =X2 f(x)=12·p+02·q = p

  • Var(X) = E(X2) –[E(X)]2 =p-p2

  • =p(1-p) = p·q

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The bernoulli distribution

例題

  • 執銅板一次,X為出現正面的數目,其分配為何?其期待值及變異數為何?

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The binomial distribution

The Binomial Distribution二項分配

定義

  • 若間斷r.v X的機率分配函數為:

  • n為完全相同且獨立之試驗的次數。

  • 每次試驗只有「成功」「失敗」兩種戶斥可能

  • p為每次試行成功之機率,失敗的機率為q = 1 – p, 其中 0<p<1。

  • 隨機變數X表示n次獨立試驗中成功之次數。

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The binomial distribution1

The Binomial Distribution二項分配

定義

  • 一個正常20歲的成年人活至65歲的機率為80%,請問三個三個年輕人中有兩人活到65歲的機率為?

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Page 236 figure 5 4

Page 236, Figure 5.4

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The binomial distribution2

The Binomial Distribution二項分配

定義

  • 每一個人存活至65的機率為85%,三人中有兩人可以存活至65:

  • (0.8)2

  • 一個死亡的機率:

  • (0.2)1

  • 根據上圖我們知道這種情形共有(ssf)(sfs)(fss)三種:

  • C3,2 = 3!/(2!(3-2)!)=3

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The binomial distribution3

The Binomial Distribution二項分配

定義

  • 鑽油井的成功機率為.30,某公司找到五處有可能蘊藏石油的地點。求正好兩處挖到石油的機率?

  • p=.3, q=1-.3=.7

  • n=5

  • 五次中有兩次成功,

  • P(正好挖到兩處)=P(X=2) = (.3)2(.7)3+ (.3)2(.7)3+…(.3)2(.7)3=10 (.3)2(.7)3=.3087

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The binomial distribution4

The Binomial Distribution二項分配

定義

  • X為五次獨立的試驗成功的次數,列出X的機率分配:

Binomial distribution, n=5 p=.3

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Page 240 figure 5 6

Page 240, Figure 5.6

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The binomial distribution5

The Binomial Distribution二項分配

  • 如果p=.5, 則成功失敗的機率各半,此機率分配為對稱(symmetric)。

  • 若p>.5,表示「成功」的機率大於「失敗」,圖形右方的機率會大於左方。

  • n愈大,機率分配愈接近鐘型(bell shaped)

  • 如果p很接近.5,既使n很小,機率分配也會呈現鐘型狀態。

  • 圖5.6顯示,隨著p增加,圖形的高峰愈往右邊偏移,且愈接近.5,愈呈現鐘型。

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The binomial distribution6

The Binomial Distribution二項分配

  • If the random variable X1, X2,…Xn form n Bernoulli trials with parameter p and if X =X1+X2…+Xn, then X has a binomial distribution with parameter n and p.

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The bernoulli distribution

例題

  • 設X~b(n,p)已知E(X)=3, Var(X)=2,求P(X=7)(中山企研)

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Cumulative binomial distribution function

Cumulative binomial distribution function

  • 累積二項分配機率函數

p =.3, n=5

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Excel

在EXCEL中求解

  • 語法:

  • BINOMDIST(成功次數number_s,實驗次數trials, 成功機率probability_s, 求累積函數cumulative)

    • Number_s   為欲求解的實驗成功次數。

    • Trials   為獨立實驗的次數。

    • Probability_s    為每一次實驗的成功機率。

    • Cumulative   為一邏輯值,主要用來決定函數的型態。如果 cumulative 為 TRUE,則 BINOMDIST 傳回累加分配函數值,其代表最多有 number_s 次成功的機率;如果其值為 FALSE,則傳回機率密度函數的機率值,代表有 number_s 次成功的機率。

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Excel1

在EXCEL中求解

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Excel2

在EXCEL中求解

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The bernoulli distribution

例題

  • According to IRS, approximately 20% of all income tax returns contain mathematical errors.

  • (a) find the probability that 3 or fewer returns out of a sample of 10 contain mathematical errors.

  • (b) Find the probability that fewer than three of the returns contain errors.

  • (c) Find the probability that exactly three of the returns contains errors.

  • (d) find the probability that three or more of the returns contain errors.

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The bernoulli distribution

例題

  • Let X denote the number of errors, then X follows the binomial distribution with n=10 and p=.20

  • (a) P(X3) = P(X=0) +P(X=1)+P(X=2)+P(X=3)

  • 查表可之n=10, p=.2, c=3 P(X 3) = .879

  • (b) P(X<3)

  • N=10, p=.2, c=2, P(X<3)= P(X 2)=.678

  • (c) P(X=3)=P(X 3) – P(X 2)=.879-.678=.201

  • (d) P(X3) =1- P(X 2) = 1-.678=.322

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The bernoulli distribution

例題

  • 生壞血病復原之機率為40%,現有15人患此病,求

  • (一)至少10人存活的機率

  • P(X10)=1-P(X9) =.0338

  • (二)3-8人存活的機率

  • P(3X 8)=P(X 8)-P(X 2)=.8779

  • (三)恰巧5人存活的機率

  • P(X=5)

  • (四)期望值及變異數

  • E(X)=15(.4)=6 Var(X)=15(.4)(.6)=3.6

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Sample proportion of successes

Sample proportion of successes

  • Statisticians frequently are more interested in the proportion of successes in a sample than in the number of successes.

  • If we obtain X successes in n trials, then the sample proportion ^p = X/n

  • P(X=x) = P(^p=x/n)

  • E(^p)=E(X/n)=np/n=p

  • The sample proportion ^p is an unbiased estimator of population proportion p.

  • Var(^p)=Var(X/n)=(1/n)2Var(X)= npq/n2 = pq/n

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The bernoulli distribution

例題

  • A councilman claims that at least 30% of the voters of a large city are in favor of increasing taxes on alcoholic beverages. To test this claim, a polling agent obtains a random sample of 500 voters. Suppose that X=100 voters in the sample say they favor the tax. Thus, the sample proportion is ^p=100/500=.2. Is it reasonable to reject the claim that, in the population, p is at least .3?

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The bernoulli distribution

例題

  • If the claim is true, the the sample proportion ^p has expected value E(^p)=p=.3

  • Var(^p)=pq/n=(.3)(.7)/500=.00042

  • S^p=sqrt(.00042)=.02

  • Empirical rule more than 99.7% of the value of ^p should fall within 3 standard deviation of the mean:

  • (.3-.06, .3+.06) = (.24, .36)

  • .02 lies outside this interval, we have strong evidence that, in the population, p does not equal to .3. If p were .3, it would be quite unusual to observe a value as extreme as ^p=.2 in a sample of n=500.

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