Random-Variables General Mathematics 11i

Random-Variables

1. Random Variables

2. Objectives: 1. Illustrate a random variable 2. Distinguish between a discrete and a continuous random variable 3. Find the probable values of a random variable 4. Illustrate a probability distribution for a discrete random variable and its properties

3. Random Variable A variable whose possible outcomes are determined by chance Typically represented by an uppercase letter, usually X

4. 2 Types of Random Variable Discrete random variables Continuous random variables

5. Discrete Random Variables A variable that can only take a finite number of distinct values Can be represented by nonnegative whole numbers An entity that have a countable value

6. Discrete Random Variables: Examples X = number of points scored by a randomly selected PBA player Y = number of birds in a nest

7. Continuous Random Variables A variable that can assume an infinite number of values in an interval between two specific values Can assume values that can be represented not only by nonnegative whole numbers but also by fractions and decimals An entity with a value that can be obtained by measuring

8. Continuous Random Variables: Examples X = the height of a randomly selected student inside the library in centimeters Y = the weights in kilogram of randomly selected dancers after taking up aerobics

9. Classify the following as discrete or continuous: 1. The number of senators present in a meeting 2. The weight of newborn babies for the month of June 3. The number of ballpens in the box

10. Classify the following as discrete or continuous: 4. The capacity of the electrical resistors 5. The amount of salt needed to bake a loaf of bread 6. The capacity of an auditorium 7. The number of households with television

11. Classify the following as discrete or continuous: 8. The height of mango trees in a farm 9. The area of lots in a subdivision 10. The number of students who joined a field trip

12. Classify the following as discrete or continuous: 11. The time it takes a student to finish a test in a particular subject 12. The number of registered nurses in a city 13. The number of winners in lotto for each month

13. Classify the following as discrete or continuous: 14. The number of stars in the night sky 15. The number of grains in a sack of rice

14. Identifying possible outcomes and values of a random variable When two fair dice are thrown simultaneously, what can be the possible outcomes? What can be the possible values?

15. Possible outcomes (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

16. Possible values The possible values are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

17. TRY THIS! The five disks with their corresponding numbers 3, 6, 5, 2 and 1 are placed in a box and mixed. Three disks at a time are then drawn out. The score is the sum of the numbers drawn. What are the possible scores? How many different scores are possible ?

18. The five disks with their corresponding numbers 3, 6, 5, 2 and 1 are placed in a box and mixed. Three disks at a time are then drawn out. The score is the sum of the numbers drawn. What are the possible scores? How many different scores are possible ? 1,2,3 - 6 1, 2, 5 -- 8 3,5,6 --14 1,2 6--9 2,3,5--10 2,3,6--11 2,5,6--13

19. ANSWER THIS! Strips of papers are placed in a box with numbers 1, 2, 5, 7, 8, and 9. Each time you draw 3 numbers from the box corresponds to a score by getting the sum of the numbers drawn.

20. ANSWER THIS! a. List down all the possible outcomes. b. What are the possible scores? c. How many different scores are possible? d. Construct a discrete probability distribution

21. Discrete Probability Distribution A listing of all possible values of a discrete random variable along with their corresponding probabilities Can be presented in tabular, graphical, or formula form

22. Discrete Probability Distribution: Properties a. The probability of each value of a discrete random variable is between 0 to 1 inclusive 0 ≤ P(x) ≤ 1 b. The sum of all probabilities is 1 ∑ P(x) = 1

23. Discrete Probability Distribution: Example When two fair dice are thrown simultaneously, the following are the possible outcomes and possible values

24. Possible outcomes (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

25. Possible values The possible values are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

26. The probabilities of each of the possible values P(x) are as follows: P(2) = P(1,1) = 1/36 P(3) = P(1,2) + P(2,1) = 2/36 = 1/18 P(4) = P(1,3) + P(2,2) + P(3,1) = 3/36 = 1/12 P(5) = P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4/36 = 1/9 P(6) = P(1,5) + P(2,4) + P(3,3) + P(4,2) + P(5,1) = 5/36

27. The probabilities of each of the possible values P(x) are as follows: P(7) = P(1,6) + P(2,5) + P(3,4) + P(4,3) + P(5,2) + P(6,1) = 6/36 = 1/6 P(8) = P(2,6) + P(3,5) + P(4,4) + P(5,3) + P(6,2) = 5/36 P(9) = P(3,6) + P(4,5) + P(5,4) + P(6,3) = 4/36 = 1/9 P(10) = P(4,6) + P(5,5) + P(6,4) = 3/36 = 1/12

28. The probabilities of each of the possible values P(x) are as follows: P(11) = P(5,6) + P(6,5) = 2/36 = 1/18 P(12) = P(6,6) = 1/36

29. The discrete probability distribution (in tabular form)

30. Question: Are the properties of a discrete probability distribution met?

31. The five disks with their corresponding numbers 3, 6, 5, 2 and 1 are placed in a box and mixed. Three disks at a time are then drawn out. The score is the sum of the numbers drawn. How many different scores are possible? What are the possible scores? Compute the mean of the possible values.