1. Chapter 5 Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance.10.20.30.40 0 1 2 3 4

2. Random Variables n A random variable is a numerical description of the outcome of an experiment. n A random variable can be classified as being either discrete or continuous depending on the numerical values it assumes. n A discrete random variable may assume either a finite number of values or an infinite sequence of values. n A continuous random variable may assume any numerical value in an interval or collection of intervals.

3. Example: JSL Appliances n Discrete random variable with a finite number of values Let x = number of TV sets sold at the store in one day where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) n Discrete random variable with an infinite sequence of values Let x = number of customers arriving in one day where x can take on the values 0, 1, 2,... where x can take on the values 0, 1, 2,... We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

4. Random Variables Question Random Variable x Type Family x = Number of dependents in Discrete size family reported on tax return Distance from x = Distance in miles from Continuous home to store home to the store site Own dog x = 1 if own no pet; Discrete or cat = 2 if own dog(s) only; = 3 if own cat(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) = 4 if own dog(s) and cat(s)

5. Discrete Probability Distributions n The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. n The probability distribution is defined by a probability function, denoted by f ( x ), which provides the probability for each value of the random variable. n The required conditions for a discrete probability function are: f ( x ) > 0 f ( x ) > 0  f ( x ) = 1  f ( x ) = 1 n We can describe a discrete probability distribution with a table, graph, or equation.

6. n Using past data on TV sales (below left), a tabular representation of the probability distribution for TV sales (below right) was developed. Number Number Units Sold of Days x f ( x ) Units Sold of Days x f ( x ) 0 80 0.40 0 80 0.40 1 50 1.25 1 50 1.25 2 40 2.20 2 40 2.20 3 10 3.05 3 10 3.05 4 20 4.10 4 20 4.10 200 1.00 200 1.00 Example: JSL Appliances

7. n Graphical Representation of the Probability Distribution.10.20.30. 40.50 0 1 2 3 4 Values of Random Variable x (TV sales) ProbabilityProbability

8. Discrete Uniform Probability Distribution n The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula. n The discrete uniform probability function is f ( x ) = 1/ N f ( x ) = 1/ Nwhere: N = the number of values the random N = the number of values the random variable may assume variable may assume n Note that the values of the random variable are equally likely.

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

10. Example: JSL Appliances n Expected Value of a Discrete Random Variable x f ( x ) xf ( x ) x f ( x ) xf ( x ) 0.40.00 0.40.00 1.25.25 1.25.25 2.20.40 2.20.40 3.05.15 3.05.15 4.10.40 4.10.40 E ( x ) = 1.20 E ( x ) = 1.20 The expected number of TV sets sold in a day is 1.2

11. n Variance and Standard Deviation of a Discrete Random Variable of a Discrete Random Variable xx -  ( x -  ) 2 f ( x )( x -  ) 2 f ( x ) 0-1.2 1.44.40.576 1-0.2 0.04.25.010 2 0.8 0.64.20.128 3 1.8 3.24.05.162 4 2.8 7.84.10.784 1.660 =   1.660 =   The variance of daily sales is 1.66 TV sets squared. The variance of daily sales is 1.66 TV sets squared. The standard deviation of sales is 1.2884 TV sets. The standard deviation of sales is 1.2884 TV sets. Example: JSL Appliances

12. n The supervisor of XYZ would like to know what the average number of absentees is daily and also what the standard deviation is of the daily employee absentee rate. Historical data provides the follow probability distribution: Daily Daily Absentees f ( x ) Absentees f ( x ) 0.50 0.50 1.23 1.23 2.12 2.12 3.10 3.10 4.02 4.02 5.02 6.01 1.00 1.00 Example: XYZ Electronics

13. n Expected Value of a Discrete Random Variable x f ( x ) xf ( x ) x f ( x ) xf ( x ) 0.50.00 0.50.00 1.23.23 1.23.23 2.12.24 2.12.24 3.10.30 3.10.30 4.02.08 4.02.08 5.02.10 6.01.06 E ( x ) = 1.01 E ( x ) = 1.01 The expected number of absentees is 1.01

14. n Variance and Standard Deviation of a Discrete Random Variable of a Discrete Random Variable xx -  ( x -  ) 2 f ( x )( x -  ) 2 f ( x ) 0-1.01 1.0201.50.5101 1-0.01 0.0001.23.0000 2 0.99 0.9801.12.1176 3 1.99 3.9601.10.3960 4 2.99 8.9401.02.1788 5 3.99 15.9201.02.3184 6 4.99 24.9001.01.2490 1.7699 =   1.7699 =   The variance of absentees is 1.7699 squared. The variance of absentees is 1.7699 squared. The standard deviation of absentees is 1.3304. The standard deviation of absentees is 1.3304. Example: XYZ Electronics