1. 1 STAT 500 – Statistics for Managers STAT 500 Statistics for Managers

2. 2 STAT 500 – Statistics for Managers Agenda for this Session Random Variables Probability Distribution of a Random Variable Parameters of a Random Variable Applications

3. 3 STAT 500 – Statistics for Managers Agenda for this Session Random Variables Probability Distribution of a Random Variable Parameters of a Random Variable Applications

4. Random Variable Definition A numerical value that depends on the outcome of a chance experiment. Type of Random Variables: Discrete versus Continuous Possible values of a random variable (Discrete) Possible values of a random variable (Continuous)

5. 5 STAT 500 – Statistics for Managers To test our understanding, let us look at the the following examples: 1.Number of defective tires in a car 2.Your body temperature 3.Number of pages in a notebook 4.The lifetime of a light bulb 5.The fuel efficiency of the car 6.The amount of annual rainfall in your neighborhood

6. 6 STAT 500 – Statistics for Managers Agenda for this Session# 3 Part 1 Random Variables Probability Distribution of a Random Variable Parameters of a Random Variable Applications

7. Probability Distribution of a Random Variable Definition The probability distribution of a discrete random variable, x gives by means of a table, formula or graph, the probability associated with each possible value of x. Example Six lots of components are ready for shipping. The number of defectives in each lot in (lot number, number of defectives) format is: (1, 0), (2, 2), (3, 0), (4, 1), (5, 2), and (6, 0). One of the lots is randomly selected and shipped to you. What is the probability distribution of the number of defectives?

8. Probability Distribution of a Random Variable The number of defective is a discrete random variable It can take three values {0, 1, 2} Let x be the number of defectives or the discrete random variable of interest P (x = 0) = P (lot # 1 or lot # 3 or lot # 6) = 3/6 P (x = 1) = P (lot # 4) = 1/6 P (x = 2) = P (lot # 2 or lot # 5) = 2/6 Probability Distribution

9. Probability Distribution of a Random Variable A consumer organization that evaluates new automobiles customarily reports the number of defectives on each car examined. Let x denote the number of major defects on a randomly selected car of a certain type. One possible distribution is: What is the probability that the number of major defects is between 2 and 5? P (x is between 2 and 5) = P(2 <= x <= 5) = P(x=2)+P(x=3)+ P(x=4)+P(x=5) = 0.829

10. 10 STAT 500 – Statistics for Managers Agenda for this Session# 3 Part 1 Random Variables Probability Distribution of a Random Variable Parameters of a Random Variable Applications

11. Parameters of a Discrete Random Variable Parameters: Mean and Variance Mean value of a discrete random variable, x is denoted by μ x or E(x). We compute this value by first, multiplying each possible value of x by the probability of observing that value and then adding the resulting quantities. μ x = E(x) = Σ x * p (x) Variance of a discrete random variable, x is denoted by σ x. σ x = = Σ (x-μ x ) 2 * p (x)

12. Parameters of a Discrete Random Variable A firm is considering two possible investments. The subjective probabilities for the five possible outcomes on returns are: Calculate the means of the two investments. Calculate the variance of the two investments Interpret or comment on the parameters

13. Parameters of a Discrete Random Variable Investment 2 will yield higher return on an average than Investment 1 But investment 2 is riskier of the two.

14. Parameters of a Discrete Random Variable A television manufacturer receives certain components in lots of three from two different suppliers. Let x and y denote the number of defective components in randomly selected lots from the first and the second suppliers, respectively. The probability distributions of a and y are as below: Compare the quality of the two suppliers

15. Parameters of a Discrete Random Variable Supplier has better quality (compare means, smaller the better) But S2 is more consistent (smaller variance)

16. 16 STAT 500 – Statistics for Managers Agenda for this Session# 3 Part 1 Random Variables Probability Distribution of a Random Variable Parameters of a Random Variable Applications

17. Application # 1 Airline Overbooking Airlines sometimes overbook flights. Suppose that for a plane with 200 seats, an airline takes 210 reservations. Define the variable x as the number of people who actually show up for a sold-out flight. From past experience, the probability distribution of x is given in the following table: What is the probability that the airline can accommodate every one who shows up for the flight? What is the probability that not all passengers can be accommodated

18. Application # 1 Airline Overbooking What is the probability that the airline can accommodate every one who shows up for the flight? What is the probability that not all passengers can be accommodated P (x <=200) = 0.82 P (x >200) = 0.18 If you are number 1 on standby, what is the probability that would make the flight? P (x <=199) = 0.64

19. Application # 2 Demand and Supply The demand for cement in Fairfax county, Virginia varies greatly from month to month. Based on past five years of data, the following probability distribution estimates monthly demand : As the manager of Home Depot, you place an order for cement equal to the mean value. What is the order quantity? When you stock as per above, what is the probability of stock out? What should be the Order Quantity, if the probability of stock out is less than 9%

20. Application # 2 Demand and Supply As the manager of Home Depot, you place an order for cement equal to the mean value. What is the order quantity? When you stock as per above, what is the probability of stock out? What should be the Order Quantity, if the probability of stock out is less than 9% Order Quantity = 200*0.1+300*0.22+.. 600*0.09 = 403 or 400 tons approx. P(stock-out) = P (D > 400) = 0.36 500

21. 21 STAT 500 – Statistics for Managers Agenda for this Session# 3 Part 1 Random Variables Probability Distribution of a Random Variable Parameters of a Random Variable Applications