3-Computing-the-Mean-of-a-Discrete-Probability-Distribution.pptx

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1. Computing the Mean of a Discrete Probability Distribution

2. Lesson Objectives At the end of this lesson, you are expected to: illustrate and calculate the mean of a discrete random variable; interpret the mean of a discrete random variable; and solve problems involving mean of probability distributions.

3. Pre-Assessment A Recap: Evaluating Summations

4. Pre-Assessment B Recap: Finding the mean of a data set.

5. Lesson Introduction Mathematicians usually consider the outcomes of a coin toss as a random event. That is, each probability of getting a head is 1 / 2 , and the probability of getting a tail is 1 / 2 . However, it is not possible to predict with 100% certainty which outcome will occur.

6. Discussion Points The mean of a discrete random variable X is the weighted average of the possible values that the random variable can take.

7. Discussion Points Illustrative Example: Consider rolling a die. What is the average number of spots that would appear?

8. Discussion Points Step 1. Construct the probability distribution for the random variable X representing the number of spots that would appear. Number of Spots X Probability P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6

9. Discussion Points Step 2. Multiply the value of the random variable X by the corresponding probability.

10. Discussion Points Step 3: Add the results obtained in Step 2. The mean tells us the average number of spots that would appear in a roll of a die. So, the average number of spots that would appear is 3.5. Although the die will never show a number, which is 3.5, this implies that rolling the die many times, the theoretical mean would be 3.5.

11. Discussion Points

12. Example 1 Grocery Items The probabilities that a customer will buy 1, 2, 3, 4, or 5 items in a grocery store are 3 /10 , 1 /10 , 1 /10 , 2 /10 , and 3 /10 , respectively. What is the average number of items that a customer will buy?

13. Solution to Example 1 Step 1. Construct the probability distribution for the random variable X representing the number of items that the customer will buy.

14. Solution to Example 1 Step 2: Multiply the value of the random variable X by the corresponding probability.

15. Solution to Example 1 Step 3: Add the results obtained in Step 2. The mean of the probability distribution is 3.1. This implies that the average number of items that the customer will buy is 3.1.

16. Example 2 Surgery Patients The probabilities that a surgeon operates on 3, 4, 5, 6, or 7 patients in any day are 0.15, 0.10, 0.20, 0.25, and 0.30, respectively. Find the average number of patients that a surgeon operates on a day.

17. Solution to Example 2 Step 1: Construct the probability distribution for the random variable X representing the number of patients that a surgeon operates on a day. Number of Patients X Probability P(X) 3 0.15 4 0.10 5 0.20 6 0.25 7 0.30

18. Solution to Example 2 Step 2: Multiply the value of the random variable X by the corresponding probability. Number of Patients X Probability P(X) X • P(X) 3 0.15 0.45 4 0.10 0.40 5 0.20 1.00 6 0.25 1.50 7 0.30 2.10

19. Solution to Example 2 Step 3: Add the results obtained in Step 2. The average number of patients that a surgeon will operate in a day is 5.45.

20. Exercise 1 Complete the table below and find the mean of the following probability distribution.

21. Exercise 2 Find the mean of the probability distribution of the random variable X, which can take only the values 1, 2, and 3, given that P(1) = 10 /33 , P(2) = 1 /3 , and P(3) = 1 2/33 .

22. Exercise 3 The probabilities of a machine manufacturing 0, 1, 2, 3, 4, or 5 defective parts in one day are 0.75, 0.17, 0.04, 0.025, 0.01, and 0.005, respectively. Find the mean of the probability distribution.

23. Exercise 4 A bakeshop owner determines the number of boxes of pandesal that are delivered each day. Find the mean of the probability distribution shown. If the manager stated that 35 boxes of pandesal were delivered in one day, do you think that this is a believable claim? Number of Boxes X Probability P(X) 35 0.10 36 0.20 37 0.30 38 0.30 39 0.10

24. Summary The mean of a discrete random variable X is the weighted average of the possible values that the random variable can take.

25. Summary To find the mean of the probability distribution , Construct the probability distribution for the random variable X Multiply the value of the random variable X by the corresponding probability. Add the results obtained in Step 2.