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Continus Probability Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continuous Probability Distributions • Uniform Probability Distribution • Normal Probability Distribution • Exponential Probability Distribution (Optional) f (x)Normal Exponential f (x) f (x)Uniform x x Business Statistics (BUSA 3101). Dr.Lari H. Arjomand x Continuous Probability Distributions n A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Data Types Data Numerical (Quantitative) Discrete Continuous Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Categorical (Qualitative) Continuous Random Variable Examples Experiment Random Variable Possible Values Weigh 100 people Weight 45.1, 78, ... Measure part life Hours 900, 875.9, ... Ask food spending Spending 54.12, 42, ... Measure time between Inter-arrival 0, 1.3, 2.78, ... arrivals time Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continuous Probability Distribution Models In this Chapter Uniform Continuous Probability Distribution Normal Exponential Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Other Continuous Probability Distributions The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2. f (x) Uniform x1 x 2 f (x) x Normal x1 x2 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand x Normal Probability Distribution • The normal probability distribution is the most important distribution for describing a continuous random variable. • It is widely used in statistical inference. f(X) Mean Median Business Statistics Mode(BUSA 3101). Dr.Lari H. Arjomand X Normal Probability Distribution It has been used in a wide variety of applications: Heights of people Scientific measurements Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution • Normal Probability Density Function 1 ( x )2 /2 2 f (x) e 2 where: = mean = standard deviation = 3.14159 e = 2.71828 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics 1- The distribution is symmetric; its skewness measure is zero. x Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics 2- The entire family of normal probability distributions is defined by its mean and its standard deviation . Standard Deviation Mean Business Statistics (BUSA 3101). Dr.Lari H. Arjomand x Normal Probability Distribution Characteristics 3- The highest point on the normal curve is at the mean, which is also the median and mode. x Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics 4- The mean can be any numerical value: negative, zero, or positive. x -10 0 20 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics 5- The standard deviation determines the width of the curve: larger values result in wider, flatter curves. = 15 = 25 x Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics 6- Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). .5 .5 x Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics #7 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean. 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean. 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Characteristics #7 99.72% 95.44% 68.26% – 3 – 1 – 2 + 3 + 1 + 2 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand x