1. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continus Probability

2. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continuous Probability Distributions Uniform Probability DistributionUniform Probability Distribution Normal Probability DistributionNormal Probability Distribution Exponential Probability Distribution (Optional)Exponential Probability Distribution (Optional) f ( x ) x x Uniform x Normal x x Exponential

3. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 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. n It is not possible to talk about the probability of the random variable assuming a particular value. n Instead, we talk about the probability of the random variable assuming a value within a given interval.

4. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Data Types

5. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continuous Random Variable Examples

6. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continuous Probability Distribution Models In this Chapter

7. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Continuous Probability Distributions n The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2. f ( x ) x x Uniform x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x Normal x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

8. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution The normal probability distribution is the most important distribution for describing a continuous random variable.The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference.It is widely used in statistical inference. Mean Median Mode

9. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Heights of people Heights Normal Probability Distribution n It has been used in a wide variety of applications: Scientific measurements measurementsScientific

10. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution Normal Probability Density FunctionNormal Probability Density Function  = mean  = standard deviation  = 3.14159 e = 2.71828 where:

11. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 1- The distribution is symmetric; its skewness 1- The distribution is symmetric; its skewness measure is zero. measure is zero. 1- The distribution is symmetric; its skewness 1- The distribution is symmetric; its skewness measure is zero. measure is zero. Normal Probability Distribution n Characteristics x

12. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 2- The entire family of normal probability 2- The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . 2- The entire family of normal probability 2- The entire family of normal probability distributions is defined by its mean  and its distributions is defined by its mean  and its standard deviation . standard deviation . Normal Probability Distribution n Characteristics Standard Deviation  Mean  x

13. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 3- The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. 3- The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. Normal Probability Distribution n Characteristics x

14. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution n Characteristics -10020 4- The mean can be any numerical value: negative, zero, or positive. zero, or positive. 4- The mean can be any numerical value: negative, zero, or positive. zero, or positive. x

15. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution n Characteristics  = 15  = 25 5- The standard deviation determines the width of the curve: larger values result in wider, flatter curves. 5- The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

16. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 6- Probabilities for the normal random variable are 6- Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). 6- Probabilities for the normal random variable are 6- Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution n Characteristics.5.5 x

17. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution n Characteristics # 7 of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.68.26%68.26% +/- 1 standard deviation of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. 95.44%95.44% +/- 2 standard deviations of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.99.72%99.72% +/- 3 standard deviations

18. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Normal Probability Distribution n Characteristics # 7 x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%