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

1. Business and Finance College Principles of StatisticsEng. Heba Hamad2008

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

3. Random Variables A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values.

4. Expected Value and Variance The expected value, or mean, of a random variable is a measure of its central location. E(x) =  = xf(x) The variance summarizes the variability in the values of a random variable. Var(x) =  2 = (x - )2f(x) The standard deviation, , is defined as the positive square root of the variance.

5. Roll of a Die f(x) = 1/6, for x = 1, 2, 3, 4, 5, 6 E(x) =  =  x*f(x) = 1*f(1) = 1 * 0.167 2*f(2) = 2 * 0.167 3*f(3) = 3 * 0.167 4*f(4) = 4 * 0.167 5*f(5) = 5 * 0.167 6*f(6) = 6 * 0.167 3.5

6. Example of DiCarlo Motors

7. Example of DiCarlo Motors • Over time DiCarlo can anticipate selling an average of 1.50 automobiles per day. • Assuming 30 days of operation during a month, we can use the expected value of 1.5 to anticipate average monthly sales of 30(1.5) = 45 automobiles.

8. Dicarlo Motors Example

9. Expected Value • Expected Value xf(x)xf(x) 0 .40 .00 1 .25 .25 2 .20 .40 3 .05 .15 4 .10 .40 E(x) = 1.20 expected number of TVs sold in a day

10. Variance • Variance and Standard Deviation x (x - )2 f(x) (x - )2f(x) x -  -1.2 -0.2 0.8 1.8 2.8 1.44 0.04 0.64 3.24 7.84 0 1 2 3 4 .40 .25 .20 .05 .10 .576 .010 .128 .162 .784 Variance of daily sales = s 2 = 1.660 Standard deviation of daily sales = 1.2884 TVs

11. Roll of a Die σ2 =  (x-  )2*f(x) (1-3.5)2*1/6 = 6.25/6 (2-3.5)2*1/6 = 2.25/6 (3-3.5)2*1/6 = 0.25/6 (4-3.5)2*1/6 = 0.25/6 (5-3.5)2*1/6 = 2.25/6 (6-3.5)2*1/6 = 6.25/6 Variance = 17.5/6 = 2.92 Standard deviation = 1.71

12. Example Find the Expected Value and the Standard Deviation

13. Example

14. Example

15. Example

16. Example