【正文】 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會計資訊系統(tǒng)計學(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 the cornerstone distribution of statistical inference. 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 813 ? A random variable X with mean m and variance s2 is normally distributed if its probability density function is given by Normal Distribution Unlike the range of the uniform distribution (a ≤ x ≤ b), Normal distributions range from minus infinity to plus infinity. ?? 2 . 7 1 8 2 8e and 3 . 1 4 1 5 9π w h e r ex eπ2σ1f ( x )2)σμx)(21(??????????2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 814 ?? 2 . 7 1 8 2 8e and 3 . 1 4 1 5 9π w h e r ex eπ2σ1f ( x )2)σμx)(21(??????????The Normal Distribution ? Important things to note: The normal distribution is fully defined by two parameters: its standard deviation and mean The normal distribution is bell shaped and symmetrical about the mean. 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 815 The Shape of the Normal Distribution The normal distribution is bell shaped, and symmetrical around m. m Why symmetrical? Let m = 100. Suppose x = 110. 22 10)2/1(100110)2/1( e21e21)1 1 0(f ?????? s??????? s???s??s?Now suppose x = 90 22 10)2/1(10090)2/1( e21e21)90(f ?????? s???????? s???s??s?110 90 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 816 The effects of m and s How does the standard deviation affect the shape of f(x)? s= 2 s =3 s =4 m = 10 m = 11 m = 12 How does the expected value affect the location of f(x)? Increasing the mean shifts the curve to the right. Increasing the standard deviation “flattens” the curve. 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 817 Standard Normal Distribution ? A normal distribution whose mean is 0 and standard deviation is 1 is called the standard normal distribution（ 標準常態(tài)分配） . ? As we shall see shortly, any normal distribution can be converted to a standard normal distribution with simple algebra. This makes calculations much easier. ?? 2 . 7 1 8 2 8e and 3 . 1 4 1 5 9π w h e r ex eπ2σ1f ( x )2)σμx)(21(??????????0 1 1 ?? and h e r ez eπ21f ( z ) 2z2?????????2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 818 ? Two facts help calculate normal probabilities: ? The normal distribution is symmetrical. ? Any normal distribution can be transformed into a specific normal distribution called “STANDARD NORMAL DISTRIBUTION” ? Example The amount of time it takes to assemble a puter is normally distributed, with a mean of 50 minutes and a standard deviation of 10 minutes. What is the probability that a puter is assembled in a time between 45 and 60 minutes? Finding Normal Probabilities XXXZsm??2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 819 XXXZsm??? Solution ? If X denotes the assembly time of a puter, we seek the probability P(45X60). ? This probability can be calculated by creating a new normal variable the standard normal variable. E(Z) = m = 0, V(Z) = s2 = 1 Every normal variable with some m and s, can be transformed into this Z. Therefore, once probabilities for Z are calculated, probabilities of any normal variable can be found. Finding Normal Probabilities This shifts the mean of X to zero… This changes the shape of the curve… 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 820 ? Example continued P(45X60) = P( ) 45 X 60 ? m 50 50 s 10 10 = P( Z 1) To plete the calculation we need to pute the probability under the standard normal distribution Finding Normal Probabilities 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 821 z 0 0. 01 ……. 0. 05 0. 060. 0 0. 00 00 0. 00 40 0. 01 99 0. 02 390. 1 0. 03 98 0. 04 38 0. 05 96 0. 06 36. . . . .. . . . .1. 0 0. 34 13 0. 34 38 0. 35 31 0. 35 54. . . . .. . . . .1. 2 0. 38 49 0. 38 69 ……. 0. 39 44 0. 39 62. . . . . .. . . . . .Standard normal probabilities have been calculated and are provided in a table . The tabulated probabilities correspond to the area between Z=0 and some Z = z0 0 Z = 0 Z = z0 P(0Zz0) Using the Standard Normal Table 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 822 ? Example continued P(45X60) = P( ) 45 X 60 ? m 50 50 s 10 10 = P( Z 1) z0 = 1 z0 = We need to find the shaded area Finding Normal Probabilities 2022會計資訊系統(tǒng)計學(xué) (一 )上課投影片 . 823 P(Z0)+ P(0Z1) P(45X60) = P(