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Chapter Five
Continuous Random Variables
Continuous Random Variables
5.1 Continuous Probability Distributions
5.2 The Uniform Distribution
5.3 The Normal Probability Distribution
*5.4 Approximating the Binomial Distribution by Using the Normal Distribution
*5.5 The Exponential Distribution
*5.6 The Cumulative Normal Table
5.1 Continuous Probability Distributions
The curve f(x) is the continuous probability distribution (or probability curve or probability density function) of
the random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x)
corresponding to the interval.
Properties of f(x)
1. f(x)  0 for all x
2. The total area under the curve of f(x) is equal to 1
5.2 The Uniform Distribution
If c and d are numbers on the real line, the probability curve describing the uniform distribution is
The mean and standard deviation of a uniform random variable x are
The Uniform Probability Curve
5.3 The Normal Probability Distribution
The normal probability distribution is defined by the equation
 and  are the mean and standard deviation,  = 3.14159 … and e = 2.71828 is the base of natural or Naperian
logarithms.
The Position and Shape of the Normal Curve
Normal Probabilities
Three Important Areas under the Normal Curve
The Empirical Rule for Normal Populations
The Standard Normal Distribution
If x is normally distributed with mean  and standard deviation , then
is normally distributed with mean 0 and standard deviation 1, a standard normal distribution.
Some Areas under the Standard Normal Curve
Calculating P(z  -1)
Calculating P(z  1)
Finding Normal Probabilities
Example 5.2
The Car Mileage Case
Procedure
1. Formulate in terms of x.
2. Restate in terms of relevant z values.
3.Find the indicated area under the standard normal curve.
Finding Z Points on a Standard Normal Curve
Finding X Points on a Normal Curve
Example 5.5 Finding the number of tapes stocked
so that P(x > st) = 0.05
Finding a Tolerance Interval
Finding a tolerance interval [  k] that contains 99% of the measurements in a normal population.
5.4 Normal Approximation to the Binomial
If x is binomial, n trials each with probability of success p and n and p are such that np  5 and n(1-p)  5, then
x is approximately normal with
Example: Normal Approximation to Binomial
Example 5.8: Approximating the binomial probability P(x = 23) by using the normal curve when
5.5 The Exponential Distribution
If l is positive, then the exponential distribution is described by the probability density function
mean mx=1/l
standard deviation sx=1/l
Example: Computing Exponential Probabilities
Given mx=3.0 or l=1/3=.333
5.6 The Cumulative Normal Table
The cumulative normal table gives of being less than or equal any given z-value
The cumulative normal table gives the shaded area
Discrete Random Variables
5.1
Continuous Probability Distributions
5.2
The Uniform Distribution
5.3
The Normal Probability Distribution
*5.4 Approximating the Binomial Distribution by Using the Normal Distribution
*5.5 The Exponential Distribution
*5.6 The Cumulative Normal Table
Chapter Five / 第五章
Continuous Random Variables / 連續的隨機變數
Continuous Random Variables / 連續的隨機變數
5.1 Continuous Probability Distributions / 5.1 連續機率分存
5.2 The Uniform Distribution / 5.2 均勻分配
5.3 The Normal Probability Distribution / 5.3 正常分佈
*5.4 Approximating the Binomial Distribution by Using the Normal Distribution / *5.4 接近二項分佈藉由使用常態
分配
*5.5 The Exponential Distribution / *5.5 個指數分配
*5.6 The Cumulative Normal Table / *5.6 個累積的常態表 /
5.1 Continuous Probability Distributions / 5.1 連續機率分存
The curve f(x) is the continuous probability distribution (or probability curve or probability density function) of the
random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x)
corresponding to the interval. / 曲線 f(x) 是隨機變數 x 的連續機率分存 (或者概率曲線或者概率密度功能) 如
果機率哪一 x 將在一個數的指定間距中是是在對間距符合的曲線 f(x) 下面的區域。
Properties of f(x) / f 的財產 (x)
1. f(x) ³ 0 for all x / 1.f(x)30 對於所有的 x
2.The total area under the curve of f(x) is equal to 1 / 2.在 f(x) 的曲線下面的總面積和 1 相等 /
5.2 The Uniform Distribution / 5.2 均勻分配
If c and d are numbers on the real line, the probability curve describing the uniform distribution is / 如果 c 和 d 在
真正的行上是數,描述均勻分配的概率曲線是
The mean and standard deviation of a uniform random variable x are / 統一隨機變數 x 的平均數和標準離差是 /
The Uniform Probability Curve / 統一的概率曲線 /
5.3 The Normal Probability Distribution / 5.3 正常分佈
The normal probability distribution is defined by the equation / 正常分佈被方程定義
and s are the mean and standard deviation, p = 3.14159 … and e = 2.71828 is the base of natural or Naperian
logarithms.
和 s 是平均數和標準離差, p=3.14159 …，而且 e=2.71828 是天然的期底或 Naperian 對數。 /
The Position and Shape of the Normal Curve / 職務和正常曲線的形成 /
Normal Probabilities / 常態機率 /
Three Important Areas under the Normal Curve / 在正常曲線下面的三個重要的區域
The Empirical Rule for Normal Populations / 為正常的人口經驗的規則 /
The Standard Normal Distribution / 標準的常態分配
If x is normally distributed with mean m and standard deviation s, then / 如果 x 是以低劣的 m 和標準離差 s 常
態分布, 然後
is normally distributed with mean 0 and standard deviation 1, a standard normal distribution. / 是以平均數常態分布
0 和標準離差 1, 一個標準的常態分配。 /
Some Areas under the Standard Normal Curve / 在標準的正常曲線下面的一些區域 /
-1) / 有心機的
-1) /
/ 有心機的
Finding Normal Probabilities / 發現正常的機率
Example 5.2 / 例子 5.2
The Car Mileage Case / 汽車運費個案 Procedure / 程序
1. Formulate in terms of x. / 1.根據 x 制定。
2. Restate in terms of relevant z values. / 2.根據有關的 z 價值重新敘述。
3. Find the indicated area under the standard normal curve. / 3.發現在標準的正常曲線下面的被指出的區域
Finding Z Points on a Standard Normal Curve / 發現 Z 在一個標準的正常曲線上指出 /
Finding X Points on a Normal Curve / 發現 X 在一個正常曲線上指出
Example 5.5 Finding the number of tapes stocked so that P(x > st) = 0.05 / 發現磁帶的數的例子 5.5 進貨了以便
P(x> st)=0.05 /
Finding a Tolerance Interval / 發現一個容限間距
Finding a tolerance interval [m ± ks] that contains 99% of the measurements in a normal population. / 發現在正常的
人口中包含 99% 的衡量的容限間距 [m ± ks] 。 /
5.4 Normal Approximation to the Binomial / 5.4 對二項式的正態近似
If x is binomial, n trials each with probability of success p and n and p are such that np ³ 5 and n(1-p) ³ 5, then / 如果 x
是二項的, n 試驗每個以成功 p 和 n 和 p 的機率是以致於 np 35 和 n(單一 p 的)35, 然後
x is approximately normal with / x 感到約計正常 /
Example: Normal Approximation to Binomial / 例子: 對二項式的正態近似
Example 5.8: Approximating the binomial probability P(x = 23) by using the normal curve when / 例子 5.8: 接近二
項式概率 P(x=23) 藉由使用正常曲線當 /
5.5 The Exponential Distribution / 5.5 指數分配
If l is positive, then the exponential distribution is described by the probability density function / 如果 l 是積極的,那
麼指數分配被概率密度功能描述
mean mx=1/l / 意謂 mx=1/l
standard deviation sx=1/l / 標準離差 sx=1/l /
Example: Computing Exponential Probabilities / 例子: 計算指數的機率 Given mx=3.0 or l=1/3=.333 / 特定的
mx=3.0 或 l=1/3=.333 /
5.6 The Cumulative Normal Table / 5.6 ，累積的常態提議
The cumulative normal table gives of being less than or equal any given z-value / 累積的正常表給在少於或者等於任
何的特定 z-價值
The cumulative normal table gives the shaded area / 累積的正常表給陰暗的區域 /
Discrete Random Variables / 間斷的隨機變數
5.1 Continuous Probability Distributions / 5.1 連續機率分存
5.2 The Uniform Distribution / 5.2 均勻分配
5.3 The Normal Probability Distribution / 5.3 正常分佈
*5.4 Approximating the Binomial Distribution by Using the Normal Distribution / *5.4 接近二項分佈藉由使用常態
分配
*5.5 The Exponential Distribution / *5.5 個指數分配
*5.6 The Cumulative Normal Table / *5.6 個累積的常態表