【正文】 Continuous Probability Distributions Chapter 8 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 82 Continuous Probability Distributions ? Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable（ 連續(xù)型隨機(jī)變數(shù)） is one that can assume an uncountable number of values in the interval (a,b). ? We cannot list the possible values because there is an infinite number of them. ? Because there is an infinite number of values, the probability that a continuous variable X will assume any particular value is 0. Why? 0 1 1/2 1/3 2/3 1/2 + 1/2 = 1 1/3 + 1/3 + 1/3 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 The probability of each value 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 83 0 1 1/2 1/3 2/3 1/2 + 1/2 = 1 1/3 + 1/3 + 1/3 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 As the number of values increases the probability of each value decreases. This is so because the sum of all the probabilities remains 1. When the number of values approaches infinity (because X is continuous) the probability of each value approaches 0. The probability of each value Continuous Probability Distributions 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 84 Point Probabilities are Zero ? Because there is an infinite number of values, the probability of each individual value is virtually 0. ? Thus, we can determine the probability of a range of values only. ? . with a discrete random variable like tossing a die, it is meaningful to talk about P(X=5), say. ? In a continuous setting (. with time as a random variable), the probability the random variable of interest, say task length, takes exactly 5 minutes is infinitesimally small, hence P(X=5) = 0. ? It is meaningful to talk about P(X ≤ 5). 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 85 ? To calculate probabilities we define a probability density function f(x). ? A function f(x) is called a probability density function（ 機(jī)率密度函數(shù)） (over the range a ≤ x ≤ b if it meets the following requirements: 1) f(x) ≥ 0 for all x between a and b, and 2) The total area under the curve between a and b is x1 x2 Area = 1 Probability Density Function ? The probability that X falls between x1 and x2 is found by calculating the area under the graph of f(x) between x1 and x2. P(x1≤X≤x2) 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 86 Uniform Distribution（ 均等分配） ? Consider the uniform probability distribution (sometimes called the rectangular probability distribution（ 矩形分配） ). ? It is described by the function: f(x) x b a 1ab1)ab(h e i g h tw i d t ha r e a???????bxa w he r e,ab 1)x(f ????ab1?2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 87 ? A random variable X is said to be uniformly distributed if its density function is ? The expected value and the variance are 12)ab()X(V2baE ( X )2????Uniform Distribution bxa w he r e,ab 1)x(f ????2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 88 ? The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons. Find the probability that sales are: ? Between 2,500 and 3,000 gallons ? More than 4,000 gallons ? Exactly 2,500 gallons Example 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 89 ? The daily sale of gasoline is uniformly distributed between 2,000 and 5,000 gallons. ? Find the probability that sales are between 2,500 and 3,000 gallons. 2022 5000 1/3000 f(x) = 1/(50002022) = 1/3000 for x: [2022,5000] x 2500 3000 P(2500?X?3000) = (30002500)(1/3000) = .1667 Example There is about a 17% chance that between 2,500 and 3,000 gallons of gas will be sold on a given day. 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 810 ? The daily sale of gasoline is uniformly distributed between 2,000 and 5,000 gallons. ? Find the probability that sales are more than 4,000 gallons. 2022 5000 1/3000 f(x) = 1/(50002022) = 1/3000 for x: [2022,5000] x 4000 P(X?4000) = (50004000)(1/3000) = .333 Example There is a 33% chance the gas station will sell more than 4,000 gallons on any given day. 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 811 ? The daily sale of gasoline is uniformly distributed between 2,000 and 5,000 gallons. ? Find the probability that sales are exactly 2,500 gallons. 2022 5000 1/3000 f(x) = 1/(50002022) = 1/3000 for x: [2022,5000] x 2500 P(X=2500) = (25002500)(1/3000) = 0 Example The probability that the gas station will sell exactly 2,500 gallons is zero. 2022會(huì)計(jì)資訊系統(tǒng)計(jì)學(xué) (一 )上課投影片 . 812 Normal Distribution（ 常態(tài)分配） ? This is the most important continuous distribution. ? Many distributions can be approximated by a normal distribution. ? The normal distribution is