Islamic University of Gaza Statistics and Probability for Engineers (ENGC 6310)

1. Islamic University of Gaza Statistics and Probability for Engineers (ENGC 6310) Lecture 3: Continuous Random Variables and Probability Distribution Prof. Dr. Yunes Mogheir Civil and Environmental Engineering Dept. First Semester/2019

4. 4-2 Probability Distributions and Probability Density Functions Figure 4-2Probability determined from the area under f(x).

5. 4-2 Probability Distributions and Probability Density Functions Definition

6. 4-2 Probability Distributions and Probability Density Functions Figure 4-3Histogram approximates a probability density function.

7. 4-2 Probability Distributions and Probability Density Functions

8. 4-2 Probability Distributions and Probability Density Functions Example

9. b) What is the probability that a metal cylinder has a diameter between 49.8 mm and 50.1 mm?

12. 4-2 Probability Distributions and Probability Density Functions Example 4-2

13. 4-2 Probability Distributions and Probability Density Functions Figure 4-5Probability density function for Example 4-2.

14. 4-2 Probability Distributions and Probability Density Functions Example 4-2 (continued)

15. 4-3 Cumulative Distribution Functions Definition

16. 4-3 Cumulative Distribution Functions Example

18. 4-3 Cumulative Distribution Functions Example 4-4

19. 4-3 Cumulative Distribution Functions Figure 4-7Cumulative distribution function for Example 4-4.

20. 4-4 Mean and Variance of a Continuous Random Variable Definition

21. 4-4 Mean and Variance of a Continuous Random Variable Example 4-6

22. 4-4 Mean and Variance of a Continuous Random Variable Example 4-8

23. Example Suppose that the diameter of a metal cylinder has a pdf of: What is the expected value of the cylinder diameter?

25. 4-5 Continuous Uniform Random Variable Definition

26. 4-5 Continuous Uniform Random Variable Figure 4-8Continuous uniform probability density function.

27. 4-5 Continuous Uniform Random Variable Mean and Variance

28. 4-5 Continuous Uniform Random Variable Example 4-9

29. 4-5 Continuous Uniform Random Variable Figure 4-9Probability for Example 4-9.

30. 4-5 Continuous Uniform Random Variable

31. 4-6 Normal Distribution Definition

32. 4-6 Normal Distribution Figure 4-10Normal probability density functions for selected values of the parameters  and 2.

33. 4-6 Normal Distribution Some useful results concerning the normal distribution

34. 4-6 Normal Distribution Definition : Standard Normal

35. 4-6 Normal Distribution Example 4-11 Figure 4-13Standard normal probability density function.

36. 4-6 Normal Distribution Standardizing

37. 4-6 Normal Distribution Example 4-13

38. 4-6 Normal Distribution Figure 4-15Standardizing a normal random variable.

39. 4-6 Normal Distribution To Calculate Probability

40. 4-6 Normal Distribution Example 4-14

41. 4-6 Normal Distribution Example 4-14 (continued)

42. 4-6 Normal Distribution Example 4-14 (continued) Figure 4-16Determining the value of x to meet a specified probability.

43. 4-7 Normal Approximation to the Binomial and Poisson Distributions • Under certain conditions, the normal distribution can be used to approximate the binomial distribution and the Poisson distribution.

44. 4-7 Normal Approximation to the Binomial and Poisson Distributions Figure 4-19Normal approximation to the binomial.

45. 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-17

46. 4-7 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Binomial Distribution

47. 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-18

48. 4-7 Normal Approximation to the Binomial and Poisson Distributions Figure 4-21Conditions for approximating hypergeometric and binomial probabilities.

49. 4-7 Normal Approximation to the Binomial and Poisson Distributions Normal Approximation to the Poisson Distribution

50. 4-7 Normal Approximation to the Binomial and Poisson Distributions Example 4-20