Random Variables

1. Random Variables

2. OBJECTIVE • Construct a probability distribution. • Find measures of center and spread for a probability distribution.

3. RELEVANCE To find the likelihood of all possible outcomes of a probability distribution and to describe the distribution.

4. Definition…… • Random Variable – a random variable, x, represents a numerical value associated with each outcome of a probability experiment; values are determined by chance (Mutually exclusive)

5. You have 4 True or False questions and you observe the number correct. Random Variable (x) # correct Possible Values of x 0, 1, 2, 3, 4 Example……

6. Count the number of siblings in your family. Random Variable (x) # of siblings Possible Values 0, 1, 2, 3, ….. Example……

7. Toss 5 coins and observe the number of heads. Random Variable (x) # of heads Possible Values 0, 1, 2, 3, 4, 5 Example……

8. 2 Types of Random Variables…… • Discrete – can be counted Ex: # of joggers • Continuous – can be measured Ex: Height, Weight, Temp, Time, Distance

9. Probability Distributions – Discrete ……

10. Probability Distribution…… • Consists of the values a random variable can assume and the corresponding probabilities of the variables. • The probabilities used are theoretical.

11. Construct a probability distribution of tossing 2 coins and getting heads. Remember: x = the number of heads possible p(x) = the probability of getting those heads Example……

12. The sample space for tossing 2 coins consists of TT HT TH HH The probability distribution of getting heads… Answer……

13. Now Graph It……

14. Construct a probability distribution of tossing 3 coins and getting heads. Remember: x = the # of heads possible P(x) = the probability of getting those heads Example……

15. The sample space of tossing 3 coins consists of HHH THH HHT THT HTH TTH HTT TTT The probability distribution…. Answer……

16. Graph It……

17. Construct a probability distribution for rolling a die. The probability distribution…. Example……

18. 1. Probability has to fall between 0 and 1 for individual probabilities 2. All the P(x)’s must add up to 1. 2 Requirements for a Probability Distribution……

19. Is it a probability distribution?Example 1…… YES

20. Example 2…… YES

21. Example 3….. NO

22. Example 4….. NO Individual P(x) does not fall between 0 and 1

23. Complete the Chart…… 1 – 0.72 = 0.28

24. Probability Functions

25. Probability Function…… • A rule that assigns probabilities to the values of the random #’s.

26. Example…… • The following function, is a probability distribution for x = 1, 2, 3, 4. Write the probability distribution.

27. Answer…… • You can set a formula in the lists using your calculator by putting x’s in L1 and setting L2 as L1/10……or you can just plug it in by hand.

28. Would the function • Be a probability distribution for x= 2, 3, 4, 5?

29. Answer…… • Here is the distribution…… • The sum of the P(x) values adds up to a number greater than 1….therefore it is NOT a probability distribution.

30. Example…… • Write the probability distribution for the function if x = 0, 1, 2

31. Mean, Standard Deviation, and Variance of a Probability Distribution Section 5.3

32. Take Note: Probability distributions may be used to represent theoretical populations, therefore, we will use population parameters and our symbols used will be for population values.

34. Example…… • A. Find the mean, variance, and standard deviation. • B. Find the probability that • C. Find the probability that

35. Mean: • We are going to do this on our graphing calculator. • Put x’s in L1 and P(x) in L2. • Set a formula in L3: L1xL2 • The sum of this column is your mean.

36. This is what it looks like on the calculator.

37. Another way to sum a list….. • 2nd Stat • Math • Sum L(#) This is actually better because you can store the mean to a location

38. You’ll need to add a L4 in your calculator. Set a formula to find the formula above. Variance:

39. The sum of L4 is your variance.

40. Standard Deviation: • The variance was 1.29.

41. Remember that Remember that Find the x’s for which you should sum their probabilities: The x’s will be between 1.9 – 2(1.14) = -0.38 1.9 + 2(1.14) = 4.18 Find the probability that

42. Remember: The x values are between -0.38 and 4.18. • All of our x’s fall in between these 2 values. • Add the probabilities that goes along with these values. • 0.2+0.1+0.3+0.4 = 1 • The answer is 1.

43. Remember that Remember that Find the x’s for which you should sum their probabilities: The x’s will be between 1.9 – 1.14 = 0.76 1.9 + 1.14 = 3.04 Find the probability that

44. Remember: The x values are between 0.76 and 3.04. • Only 3 of the x values fall between the values listed above. • Add the probabilities that go with those 3. • 0.1+0.3+0.4 = 0.8 • The answer is 0.8.

45. Example…… • A. Set up the distribution for • B. Find the mean, variance, and standard deviation of the probability distribution. • C. Find the probability that

46. Set up the distribution……

47. Mean: • We are going to do this on our graphing calculator. • Put x’s in L1 and P(x) in L2. • Set a formula in L3: L1xL2 • The sum of this column is your mean.

48. This is what it looks like on the calculator……

49. You’ll need to add a L4 in your calculator. Set a formula to find the formula above. The sum of list 4 is the variance. Variance:

50. Standard Deviation: • The variance was 0.555555555. • The standard deviation is 0.75.