1. Business and Finance College Principles of Statistics Eng. Heba Hamad 2008

2. Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

3. Continuous Probability Distributions Chapter 6

4. Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f ( x ) x x Uniform x Normal x x Exponential

5. Continuous Probability Distributions 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.

6. 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 x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

7. Uniform Probability Distribution where: a = smallest value the variable can assume b = largest value the variable can assume b = largest value the variable can assume f ( x ) = 1/( b – a ) for a < x < b f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 0 elsewhere A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. The uniform probability density function is:

8. Var( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 Uniform Probability Distribution Expected Value of x Variance of x

9. Uniform Probability Distribution Example: Slater's Buffet Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.

10. n Uniform Probability Density Function f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewhere where: x = salad plate filling weight x = salad plate filling weight Uniform Probability Distribution

11. n Expected Value of x n Variance of x E( x ) = ( a + b )/2 E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = (15 – 5) 2 /12 = 8.33 = 8.33 Uniform Probability Distribution

12. for Salad Plate Filling Weight f(x)f(x) f(x)f(x) x x 5 5 10 15 1/10 Salad Weight (oz.) Uniform Probability Distribution

13. f(x)f(x) f(x)f(x) x x 5 5 10 15 1/10 Salad Weight (oz.) P(12 < x < 15) = 1/10(3) =.3 What is the probability that a customer What is the probability that a customer will take between 12 and 15 ounces of salad? will take between 12 and 15 ounces of salad? 12 Uniform Probability Distribution

14. 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.

15. Heights of people Normal Probability Distribution n It has been used in a wide variety of applications: Scientific measurements measurements

16. Amounts of rainfall Normal Probability Distribution n It has been used in a wide variety of applications: Test scores scores

17. Normal Probability Distribution Normal Probability Density Function  = mean  = standard deviation  = 3.14159 e = 2.71828 where:

18. The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. Normal Probability Distribution n Characteristics x

19. The entire family of normal probability 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

20. The highest point on the normal curve is at the 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

21. Normal Probability Distribution n Characteristics -10020 The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. x

22. Normal Probability Distribution n Characteristics  = 15  = 25 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

23. Probabilities for the normal random variable are 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

24. Normal Probability Distribution n Characteristics 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% +/- 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. 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. 99.72% +/- 3 standard deviations

25. Normal Probability Distribution n Characteristics x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%