Lesson6 Mathmatical Expectations, mean of rv.pptx

son6 Mathmatical Expectations, mean of rv.pptx

1. TAIBAH UNIVERSITY Faculty of Science Department of Math. جامعة طيبة كلية العلوم قسم الرياضيات Probability and Statistics for Engineers STAT 301 Teacher : Osama Hosam Second Semester 1432/1433

2. Mathematical Expectation Lesson 6

3. Mean of a Random Variable

4. Mean of a Random Variable Definition 4.1: Let X be a random variable with a probability distribution f(x) . The mean (or expected value) of X is denoted by (or E(X) ) and is defined by:

5. Mean of a Random Variable (Example1) A shipment of 8 similar microcomputers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the expected number of defective computers purchased

6. The prob. distribution of X is (see Lesson 15) x 1 2 Total f(x)= P(X=x) Hypergeometric Distribution 1.00 Mean of a Random Variable (Example1)

7. The expected value of the number of defective computers purchased is the mean (or the expected value) of X , which is: (computers) Mean of a Random Variable (Example1)

8. Mean of a Random Variable (Ex 4.3) Let X be a continuous random variable that represents the life (in hours) of a certain electronic device. The pdf of X is given by: Find the expected life of this type of devices.

9. Solution: (hours) Therefore, we expect this type of electronic devices to last, on average, 200 hours.

10. Mean of a Random Variable Let X be a random variable with a prob. distribution f(x ) , and let g(X ) be a function of the random variable X . The mean (or expected value) of the random variable g(X ) is denoted by (or ) and is defined by: Theorem 4.1:

11. Mean of a Random Variable (Example 3) Let X be a discrete random variable with the following probability distribution x 1 2 f(x) Find:

12. Mean of a Random Variable (Example 3) Solution

13. Mean of a Random Variable (Example4) Let X be a continuous random variable that represents the life (in hours) of a certain electronic device. The pdf of X is given by: Find

14. Mean of a Random Variable (Example 4) Solution

15. Mean of a Random Variable (Ex. 4.5) Let X be a random variable with density function Find

16. Mean of a Random Variable (Ex4.5) Solution

17. Exercises

18. Variance

19. Variance of a Random Variable In Figure 4.1 we have the histograms of two discrete probability distributions with the same mean  = 2 that differ considerably in the variability or dispersion of their observations about the mean. Figure 4.1 Distributions with equal means and unequal dispersions.

20. Variance of a Random Variables The most important measure of variability of a random variable X is called the variance of X and is denoted by Var (X) or

21. Variance of a Random Variables Let X be a random variable with a probability distribution f(x) and mean  . The variance of X is defined by: Definition 4.3:

22. Variance of a Random Variable Definition : The positive square root of the variance of X , ,is called the standard deviation of X Note:

23. Variance of a Random Variable (Ex 4.8) Let the random variable X represent the number of automobiles that are used for official business purposes on any given workday. The probability distribution for company A [ Figure 4.1(a )] is x 1 2 3 f(x) 0.3 0.4 0.3

24. Variance of a Random Variables(Ex 4.8) For company B [ Figure 4.1(b )] is x 1 2 3 4 f(x) 0.2 0.1 0.3 0.3 0.1 Show that the variance of the probability dist. for company B is greater than that of company A.

25. Variance of a Random Variables(Ex 4.8) Solution: For company A we find that:

26. Variance of a Random Variables(Ex 4.8) For company B we find that:

27. Variance of a Random Variables(Ex 4.8) Clearly, the variance of the number of automobiles that are used for official business purposes is greater for company B than for company A.

28. Variance of a Random Variables The variance of a random variable X is Theorem 4.2:

29. Variance of a Random Variable For the discrete case we can write: Proof:

30. Let X be a discrete random variable with the following probability distribution x 1 2 3 f(x) 0.15 0.38 0.10 0.01 Find Var (X) . Variance of a Random Variables(Ex 4.9)

31. Variance of a Random Variables(Ex 4.8) Solution: =0.87  (0.61) 2 = 0.4979

32. The weekly demand for Pepsi, in thousands of liters , from a local chain of efficiency stores, is a continuous random variable X having the probability density Find the mean and variance of X . Variance of a Random Variable (Ex 4.10)

33. Variance of a Random Variables(Ex 4.10) Solution:

34. Mean of a Random Variable Let X be a random variable with a prob. distribution f(x ), and let g(X ) be a function of the random variable X . The variance of the random variable g(X ) is : Theorem 4.3:

35. Calculate the variance of g(X) = 2X +3 where X be a discrete random variable with the following probability distribution x 1 2 3 f(x) 0.25 0.125 0.5 0.125 Variance of a Random Variables(Ex 4.11)

36. Solution Variance of a Random Variables(Ex 4.11) 2x+3 3 5 7 9 f(x) 0.25 0.125 0.5 0.125 First let us find the mean of the random variable 2X +3

37. Variance of a Random Variables(Ex 4.11) 4x 2 -12x+9 9 1 1 9 f(x) 0.25 0.125 0.5 0.125

38. Variance of a Random Variable If we use Theorem 4.2 we will get the same answer:

39. Variance of a Random Variables(Ex 4.11) 4x 2 +12x+9 9 25 49 81 f(x) 0.25 0.125 0.5 0.125

40. Exercise

41. Means and Variances of Linear Combinations of Random Variables

42. If X 1 , X 2 , …, X n are n random variables and a 1 , a 2 , …, a n are constants, then the random variable : is called a linear combination of the random variables X 1 ,X 2 ,…, X n . Linear Combination of random variables

43. If X is a random variable with mean  =E(X ), and if a and b are constants, then: E( aXb ) = a E(X)  b     aXb = a  X ± b Theorem 4.5: Linear Combination of random variables

44. Setting a = 0 in Theorem 4.5 , we see that E(b) = b . Corollary 4.1: Corollary 4.2: Setting b = 0 in Theorem 4.5 , we see that E( aX ) = aE (X) . Linear Combination of random variables

45. Let X be a random variable with the following probability density function: Find E(4X+3). Linear Combination of r.v . Example 4.16

46. Solution: 5/4 Linear Combination of r.v . Example 4.16 E(4X+3) = 4 E(X)+3 = 4(5/4) + 3 =8

47. Linear Combination of r.v . Example 4.16 Another solution: ; g(X) = 4X+3 E(4 X +3) =

48. If X 1 , X 2 , …, X n are n random variables and a 1 , a 2 , …, a n are constants, then: E(a 1 X 1 +a 2 X 2 + … + a n X n ) = a 1 E(X 1 )+ a 2 E(X 2 )+ …+ a n E ( X n )  Theorem : Linear Combination of random variables

49. Corollary : Linear Combination of random variables If X , and Y are random variables, then: E(X ± Y) = E(X) ± E(Y)

50. Linear Combination of random variables Theorem 4.9 : If X is a random variable with variance and if a and b are constants, then: 

51. Setting a = 1 in Theorem 4.9 , we see that Corollary 4.6: Corollary 4.7: Setting b = 0 in Theorem 4.9 , we see that Linear Combination of random variables

52. If X 1 , X 2 , …, X n are n independent random variables and a 1 , a 2 , …, a n are constants, then:  Theorem : Linear Combination of random variables

53. If X, and Y are independent random variables, then:·    Corollary : Linear Combination of random variables

54. Linear Combination of r.v . (EX 4.20) If X, and Y are independent random variables, with .    Find the variance of the random variable

55. Solution: Linear Combination of r.v . (EX 4.20)

56. If X, and Y are independent random variables, with .    Linear Combination of r.v . (Example) Find:

57. Solution: Linear Combination of r.v . (EX 4.20)

58. Exercise

59. The End