Chapter 6 Continuous Probability Distributions

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Chapter 6Chapter 6 Continuous Probability Distributions Continuous Probability Distributions

Uniform Probability DistributionUniform Probability Distribution Normal Probability DistributionNormal Probability Distribution Exponential Probability DistributionExponential Probability Distribution

xx

ff((xx))

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Continuous Probability DistributionsContinuous Probability Distributions

A A continuous random variablecontinuous random variable can assume any can assume any value in an interval on the real line or in a value in an interval on the real line or in a collection of intervals.collection of intervals.

It is not possible to talk about the probability of It is not possible to talk about the probability of the random variable assuming a particular value.the random variable assuming a particular value.

Instead, we talk about the probability of the Instead, we talk about the probability of the random variable assuming a value within a given random variable assuming a value within a given interval.interval.

The probability of the random variable assuming The probability of the random variable assuming a value within some given interval from a value within some given interval from xx11 to to xx22 is defined to be the is defined to be the area under the grapharea under the graph of the of the probability density functionprobability density function （概率密度函（概率密度函数）数） betweenbetween x x11 andand x x22..

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A random variable is A random variable is uniformly distributeduniformly distributed whenever the probability is proportional to the whenever the probability is proportional to the interval’s length. interval’s length.

Uniform Probability Density FunctionUniform Probability Density Function

ff((xx) = 1/() = 1/(bb - - aa) for ) for aa << xx << bb

= 0 = 0 elsewhere elsewhere

where: where: aa = smallest value the variable can = smallest value the variable can assumeassume

bb = largest value the variable can = largest value the variable can assumeassume

Uniform Probability Distribution Uniform Probability Distribution （均匀概率分布）（均匀概率分布）

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Uniform Probability DistributionUniform Probability Distribution

Expected Value of Expected Value of xx

E(E(xx) = () = (aa + + bb)/2)/2

Variance of Variance of xx

Var(Var(xx) = () = (bb - - aa))22/12/12

where: where: aa = smallest value the variable can = smallest value the variable can assumeassume

bb = largest value the variable can = largest value the variable can assumeassume

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Example: Slater's BuffetExample: Slater's Buffet


Slater customers are charged for the Slater customers are charged for the amount of salad they take. Sampling suggests amount of salad they take. Sampling suggests that the amount of salad taken is uniformly that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.distributed between 5 ounces and 15 ounces.

The probability density function isThe probability density function is

ff((xx) = 1/10 for 5 ) = 1/10 for 5 << xx << 15 15

= 0 = 0 elsewhere elsewhere

where:where:

xx = salad plate filling weight = salad plate filling weight

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What is the probability that a customer What is the probability that a customer will will take between 12 and 15 ounces of take between 12 and 15 ounces of salad?salad?

f(x)f(x)

x x55 1010 15151212

1/101/10

Salad Weight (oz.)Salad Weight (oz.)

P(12 < x < 15) = 1/10(3) = .3P(12 < x < 15) = 1/10(3) = .3

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Expected Value of Expected Value of xx

E(E(xx) = () = (aa + + bb)/2)/2

= (5 + 15)/2= (5 + 15)/2

= 10= 10 Variance of Variance of xx

Var(Var(xx) = () = (bb - - aa))22/12/12

= (15 – 5)= (15 – 5)22/12/12

= 8.33= 8.33

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Normal Probability DistributionNormal Probability Distribution （正态概率分布）（正态概率分布）

Graph of the Normal Probability Density Graph of the Normal Probability Density FunctionFunction

xx

ff((xx))

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Normal Probability DistributionNormal Probability Distribution

Characteristics of the Normal Probability Characteristics of the Normal Probability DistributionDistribution

• The shape of the normal curve is often The shape of the normal curve is often illustrated as a illustrated as a bell-shaped curvebell-shaped curve. .

• Two parametersTwo parameters, , (mean) and (mean) and (standard (standard deviation), determine the location and shape deviation), determine the location and shape of the distribution.of the distribution.

• The The highest pointhighest point on the normal curve is at on the normal curve is at the mean, which is also the median and the mean, which is also the median and mode.mode.

• The mean can be any numerical value: The mean can be any numerical value: negative, zero, or positive.negative, zero, or positive.

… continued

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Characteristics of the Normal Probability Characteristics of the Normal Probability DistributionDistribution

• The normal curve is The normal curve is symmetricsymmetric..

• The standard deviation determines the The standard deviation determines the width of the curve: larger values result in width of the curve: larger values result in wider, flatter curves.wider, flatter curves.

• The total area under the curve is 1 (.5 to the The total area under the curve is 1 (.5 to the left of the mean and .5 to the right).left of the mean and .5 to the right).

• Probabilities for the normal random variable Probabilities for the normal random variable are given by are given by areas under the curveareas under the curve..

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% of Values in Some Commonly Used Intervals% of Values in Some Commonly Used Intervals

• 68.26%68.26% of values of a normal random of values of a normal random variable are within variable are within +/- 1+/- 1 standard standard deviationdeviation of its mean. of its mean.

• 95.44%95.44% of values of a normal random of values of a normal random variable are within variable are within +/- 2+/- 2 standard standard deviationsdeviations of its mean. of its mean.

• 99.72%99.72% of values of a normal random of values of a normal random variable are within variable are within +/- 3+/- 3 standard standard deviationsdeviations of its mean. of its mean.

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Normal Probability Density FunctionNormal Probability Density Function

where:where:

= mean= mean

= standard deviation= standard deviation

= 3.14159= 3.14159

ee = 2.71828 = 2.71828

f x e x( ) ( ) / 12

2 2 2

f x e x( ) ( ) / 1

2

2 2 2

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Standard Normal Probability DistributionStandard Normal Probability Distribution

A random variable that has a normal A random variable that has a normal distribution with a mean of zero and a standard distribution with a mean of zero and a standard deviation of one is said to have a deviation of one is said to have a standard standard normal probability distributionnormal probability distribution..

The letter The letter z z is commonly used to designate this is commonly used to designate this normal random variable.normal random variable.

Converting to the Standard Normal DistributionConverting to the Standard Normal Distribution

We can think of We can think of zz as a measure of the number as a measure of the number of standard deviations of standard deviations xx is from is from ..

zx

zx

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Example: GrearExample: Grear 轮胎公司问题轮胎公司问题

Standard Normal Probability DistributionStandard Normal Probability Distribution

GrearGrear 公司刚刚开发了一种新的轮胎，并通过公司刚刚开发了一种新的轮胎，并通过一家全国连锁的折扣商店出售。因为该轮胎是一种新一家全国连锁的折扣商店出售。因为该轮胎是一种新产品，产品， GrearGrear 公司的经理们认为是否保证一定的行驶公司的经理们认为是否保证一定的行驶里程数将是该产品能否被顾客接受的重要因素。在制里程数将是该产品能否被顾客接受的重要因素。在制定这种轮胎的里程质保政策之前，经理们需要知道轮定这种轮胎的里程质保政策之前，经理们需要知道轮胎行驶里程数的概率信息。胎行驶里程数的概率信息。

根据对这种轮胎的实际路面测试，公司的工程根据对这种轮胎的实际路面测试，公司的工程师小组估计它们的平均行驶里程为师小组估计它们的平均行驶里程为 3650036500 英里，里程英里，里程数的标准差为数的标准差为 50005000 。另外，收集到的数据显示，行。另外，收集到的数据显示，行驶里程数符合正态分布应该是一个合理的假设。驶里程数符合正态分布应该是一个合理的假设。

问题是有多大百分比的轮胎能够行驶超过问题是有多大百分比的轮胎能够行驶超过4000040000 英里？换句话说，轮胎行驶里程大于英里？换句话说，轮胎行驶里程大于 4000040000英里的概率是多少？英里的概率是多少？

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PP （（ x>=40000x>=40000 ）） =0.5-0.258=0.242=0.5-0.258=0.242说明：大约有说明：大约有 24.2%24.2% 的轮胎行使里程会超过的轮胎行使里程会超过 4000040000 。。

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现在我们假设公司正在考虑一项质量保政策，如果初现在我们假设公司正在考虑一项质量保政策，如果初始购买的轮胎没有能够使用到保证的里程数，公司将始购买的轮胎没有能够使用到保证的里程数，公司将以折扣价格为客户更换轮胎。如果公司希望符合折扣以折扣价格为客户更换轮胎。如果公司希望符合折扣条件的轮胎不超过条件的轮胎不超过 10%10% ，则保证的里程应为多少？，则保证的里程应为多少？


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分析分析 11 ：处在均值和未知保证里程数之间的面积必须：处在均值和未知保证里程数之间的面积必须为为 40%40% 。。


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分析分析 22 ：在表中查找：在表中查找 0.40.4 ，看到该面积大约在均值与，看到该面积大约在均值与小于均值小于均值 1.281.28 个标准差处之间，即个标准差处之间，即 z=-1.28z=-1.28 是对应是对应于公司在正态分布中保证里程数的标准正态分布。于公司在正态分布中保证里程数的标准正态分布。


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为了得到对应于为了得到对应于 z=-1.28z=-1.28 的里程数的里程数 xx ，我们有：，我们有：

因此，因此， 3010030100 英时的质量保证将满足只有大约英时的质量保证将满足只有大约 10%10%的轮胎需要折价更换的要求。也许，根据这一信息，的轮胎需要折价更换的要求。也许，根据这一信息，公司将把它的轮胎里程保证设在公司将把它的轮胎里程保证设在 3000030000 英里。英里。


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Exponential Probability DistributionExponential Probability Distribution指数概率分布指数概率分布

Exponential Probability Density FunctionExponential Probability Density Function

for for xx >> 0, 0, > 0 > 0

where: where: = mean = mean

ee = 2.71828 = 2.71828

f x e x( ) / 1

f x e x( ) / 1

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Exponential Probability DistributionExponential Probability Distribution （指数概率分（指数概率分布）布）

Cumulative Exponential Distribution FunctionCumulative Exponential Distribution Function

where:where:

xx00 = some specific value of = some specific value of xx

P x x e x( ) / 0 1 o P x x e x( ) / 0 1 o

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Exponential Probability DistributionExponential Probability Distribution

The time between arrivals of cars at Al’s The time between arrivals of cars at Al’s Carwash follows an exponential probability Carwash follows an exponential probability distribution with a mean time between arrivals distribution with a mean time between arrivals of 3 minutes. Al would like to know the of 3 minutes. Al would like to know the probability that the time between two probability that the time between two successive arrivals will be 2 minutes or less.successive arrivals will be 2 minutes or less.

PP((xx << 2) = 1 - 2.71828 2) = 1 - 2.71828-2/3-2/3 = 1 - .5134 = 1 - .5134 = .4866= .4866

Example: Al’s CarwashExample: Al’s Carwash

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Example: Al’s CarwashExample: Al’s Carwash

Graph of the Probability Density FunctionGraph of the Probability Density Function

xx

f(x)f(x)

.1.1

.3.3

.4.4

.2.2

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

P(x < 2) = area = .4866P(x < 2) = area = .4866

Time Between Successive Arrivals (mins.)Time Between Successive Arrivals (mins.)

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Relationship between the PoissonRelationship between the Poissonand Exponential Distributionsand Exponential Distributions

(If) the Poisson distribution(If) the Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

(If) the Poisson distribution(If) the Poisson distributionprovides an appropriate descriptionprovides an appropriate description

of the number of occurrencesof the number of occurrencesper intervalper interval

(If) the exponential distribution(If) the exponential distributionprovides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences

(If) the exponential distribution(If) the exponential distributionprovides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences

Chapter 6 Continuous Probability Distributions

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