DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 6 Spring 2008 Chapter 6: Continuous Probability Distribution 2 � When a RV x is discrete, we can assign a positive probability to each value that x can take and get the probability distribution for x. � The sum of all probabilities of all values of x is 1. Th f ll b biliti f ll l f i 1 � Not all experiments result in RVs that are discrete. � Continuous RV, such as heights, weights, lifetime of a particular product, experimental laboratory error…etc. can assume infinitely many values corresponding to points on line interval 1 QM-120, M. Zainal
Chapter 6: Continuous Probability Distribution 3 � Suppose we have a set of measurements on a continuous RV, and we want to create a relative frequency histogram to describe their distribution. � For a small number of measurements, we can use small F ll b f t ll number of classes Chapter 6: Continuous Probability Distribution 4 � For a larger number of measurements, we must use a larger number of classes and reduce the class width 2 QM-120, M. Zainal
Chapter 6: Continuous Probability Distribution 5 � As the number of measurements become very large, the class width become very narrow, the relative frequency histogram appears more and more like a smooth curve. � This smooth curve describes the probability distribution of a continuous random variable. Chapter 6: Continuous Probability Distribution 6 � A continuous RV can take on any of an infinite number of values on the real line � The depth or density of the probability, which varies with x, maybe described by a mathematical formula f(x), called the probability density function for the RV x. 3 QM-120, M. Zainal
Chapter 6: Continuous Probability Distribution 7 � The probability distribution f(x); P(a< x <b) is equal to the shaded area under the curve � Several important properties of continuous probability distribution parallel their discrete part. Chapter 6: Continuous Probability Distribution 8 � Just as the sum of discrete probabilities is equal to 1, i.e. Σ P(x) = 1, and the probability that x falls into a certain interval can be found by summing the probabilities in that interval, continuous probability distributions have the interval, continuous probability distributions have the following two characteristics: � The area under a continuous probability distribution is equal to 1. � The probability that x will fall into a particular interval ‐ say, from a to b – is equal to the area under the curve between the two points a and b the two points a and b 4 QM-120, M. Zainal
Chapter 6: Continuous Probability Distribution 9 � There is a very important difference between the continuous case and the discrete one which is: � P(x = a) = 0 � P(a < x < b) = P(a ≤ x ≤ b) � P( x < a) = P( x ≤ a) Chapter 6: The Normal Distribution 10 � Among many continuous probability distributions, the normal distribution is the most important and widely used one. � A large number of real world phenomena are either exactly or A l b f l ld h ith tl approximately normally distributed. � The normal distribution or Gaussian distribution is given by a bell ‐ shaped curve. � A continuous RV x that has a normal distribution is said to be a normal RV with a mean μ and a standard deviation σ or simply x~N( μ , σ ). 5 QM-120, M. Zainal
Chapter 6: The Normal Distribution 11 � Note that not all the bell ‐ shaped curves represent a normal distribution Chapter 6: The Normal Distribution 12 � The normal probability distribution, when plotted, gives a bell ‐ shaped curve such that � The total area under the curve is 1.0. � The curve is symmetric around the mean. � The two tails of the curve extended indefinitely. � The mean μ and the standard deviation σ are the parameters of the normal distribution. Given the values of these two parameters, we can find the area under the normal curve for any interval. 6 QM-120, M. Zainal
Chapter 6: The Normal Distribution 13 � There is not just one normal distribution curve but rather a family of normal distribution. � Each different set of values of μ and σ gives different normal curve. � The value of μ determines � the center of the curve on the � Horizontal axis and the value � of σ gives the spread of the � normal curve Chapter 6: The Normal Distribution 14 � Like all other distributions, the normal probability distribution can be expressed by a mathematical function 2 ⎡ − μ ⎤ 1 x − 1 ⎢ ⎥ = σ ⎣ ⎦ 2 f x ( ) e σ π 2 � in which the probability that x falls between a and b is the integral of the above function from a to b, i.e. 2 2 ⎡ ⎡ − μ ⎤ ⎤ b 1 x = ∫ − 1 ⎢ ⎥ < < ⎣ σ ⎦ 2 P a ( x b ) e dx σ π 2 a 7 QM-120, M. Zainal
Chapter 6: The Normal Distribution 15 � However we will not use the formula to find the probability. instead, we will use Table VII of Appendix C. Chapter 6: The Standard Normal Distribution 16 � The standard normal distribution is a special case of the normal distribution where the μ is zero and the σ is 1. � The RV that possesses the standard normal distribution is denoted by z and it is called z values or z scores. μ = 0 σ = 1 -4 -2 0 2 4 z 8 QM-120, M. Zainal
Chapter 6: The Standard Normal Distribution 17 � Since μ is zero and the σ is 1 for the standard normal, a specific value of z gives the distance between the mean and the point represented by z in terms of the standard deviation. � The z values to the right side of the mean are positive and those on the left are negative BUT the area under the curve is always positive. � For a value of z = 2, we are 2 standard deviations from the mean (to the right). mean (to the right). � Similarly, for z = ‐ 2, we are 2 standard deviations from the mean (to the left) Chapter 6: The Standard Normal Distribution 18 9 QM-120, M. Zainal
Chapter 6: The Standard Normal Distribution 19 � Table VII of Appendix C, list the area under the standard normal curve between z = 0 to the values of z to 3.09. � To read the standard normal table we always start at z = 0 � To read the standard normal table, we always start at z 0, which represents the mean. � Always remember that the normal curve is symmetric, that is the area from 0 to any positive z value is equal to the area from 0 to that value in the negative side. Example: Find the area under the standard normal curve between z = 0 to z = 1.95 Chapter 6: The Standard Normal Distribution 20 Example: Find the area under the standard normal curve between z = ‐ 2.17 to z = 0 10 QM-120, M. Zainal
Chapter 6: The Standard Normal Distribution 21 Example: Find the following areas under the standard normal curve. a) Area to the right of z = 2.32 b) Area to the left of z = ‐ 1.54 Chapter 6: The Standard Normal Distribution 22 Example: Find the following probabilities for the standard normal curve. a) P(1.19 < z < 2.12) ) ( ) b) P( ‐ 1.56 < z < 2.31) c) P(z > ‐ .75) 11 QM-120, M. Zainal
Chapter 6: The Standard Normal Distribution 23 � Remember: When two points are on the same side of the mean first When two points are on the same side of the mean, first find the areas between the mean and each of the two points. Then, subtract the smaller area from the larger area When two points are on different side of the mean, first find the areas between the mean and each of the two points. Then, add these two area Chapter 6: The Standard Normal Distribution 24 � Empirical rule 12 QM-120, M. Zainal
Chapter 6: The Standard Normal Distribution 25 Example: Find the following probabilities for the standard normal curve. a) P(0 < z < 5.65) b) P( z < ‐ 5.3) Chapter 6: Standardizing a Normal Distribution 26 � As it was shown in the previous section, Table VII of Appendix C can be used only to find the areas under the standard normal curve. � However, in real ‐ world applications, most of continuous RVs that are normally distributed come with mean and standard deviation different from 0 and 1, respectively. � What shall we do? Is there any way to bring μ to zero and σ to 1. to 1. � Yes, it can be done by subtracting μ from x and dividing the result by σ (standardizing ) 13 QM-120, M. Zainal
Chapter 6: Standardizing a Normal Distribution 27 � Standardizing x: For a normal RV x with mean μ and standard deviation σ . The standardized RV z can be found using the following f formula l − μ x = z σ Chapter 6: Standardizing a Normal Distribution 28 Example: For the following data [3.2, 1.9, 2.7, 4.3, 3.5, 2.8, 3.9, 4.4, 1.5, 1.8], a) find the mean and standard deviation b) b) standardize each value in the sample d d h l h l x Transformation z 3.2 1.9 2.7 n ∑ ∑ 4.3 X X i 3.5 μ = = = i 1 3 2.8 n 3.9 σ = 0.99 4.4 1.5 1.8 14 QM-120, M. Zainal
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