1. Discrete Probability Distributions

2. Random Variable Random variable is a variable whose value is subject to variations due to chance. A random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability.variableprobability

3. Discrete Random Variable

4. Continuous Random Variable

5. Discrete Random Variables

6. Discrete Probability Distribution

8. Discrete Random Variable Summary Measures Expected Value : the expected value of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average correspond to the probabilities in case of a discrete random variable, or densities in case of a continuous random variable ;random variableweighted averageprobabilitiesdensities

9. Discrete Random Variable Summary Measures Standard deviation shows how much variation or "dispersion" exists from the average (mean, or expected value);dispersionmean

10. Discrete Random Variable Summary Measures

11. Probability Distributions

12. The Bernoulli Distribution Bernoulli distribution, is a discrete probability distribution, which takes value 1 with success probability p and value 0 with failure probability q=1-p.discreteprobability distribution The Probability Function of this distribution is; The Bernoulli distribution is simply Binomial (1,p).

13. Bernoulli Distribution Characteristics

14. The Binomial Distribution

15. Counting Rule for Combinations

16. Binomial Distribution Formula

17. Binomial Distribution

18. Binomial Distribution Characteristics

19. Binomial Characteristics

20. Binomial Distribution Example

21. Geometric Distribution The geometric distribution is either of two discrete probability distributions:discrete probability distributions The probability distribution of the number of X Bernoulli trials needed to get one success, supported on the set { 1, 2, 3,...}Bernoulli trials The probability distribution of the number Y = X − 1 of failures before the first success, supported on the set { 0, 1, 2, 3,... }

22. It’s the probability that the first occurrence of success require k number of independent trials, each with success probability p. If the probability of success on each trial is p, then the probability that the kth trial (out of k trials) is the first success is The above form of geometric distribution is used for modeling the number of trials until the first success. By contrast, the following form of geometric distribution is used for modeling number of failures until the first success: Geometric Distribution

23. Geometric Distribution Characteristics

24. The Poisson Distribution

25. Poisson Distribution Formula

26. Poisson Distribution Characteristics

27. Graph of Poisson Probabilities

28. Poisson Distribution Shape

29. The Hypergeometric Distribution

30. Hypergeometric Distribution Formula

31. Hypergeometric Distribution Example

32. Continuous Probability Distributions

34. The Normal Distribution

35. Many Normal Distributions

36. The Normal Distribution Shape

37. Finding Normal Probabilities

38. Probability as Area Under the Curve

39. Empirical Rules

40. The Empirical Rule

41. Importance of the Rule

42. The Standart Normal Distribution

43. The Standart Normal

44. Translation to the Standart Normal Distribution

45. Example

46. Comparing x and z units

47. The Standart Normal Table

49. General Procedure for Finding Probabilities

50. z Table Example

52. Solution : Finding P(0 < z <0.12)

53. Finding Normal Probabilities

55. Upper Tail Probabilities

57. Lower Tail Probabilities

59. The Uniform Distribution

61. The Mean and the Standart Deviation for Uniform Distribution

62. The Uniform Distribution

63. Characteristics;

64. The Exponential Distribution

66. Shape of the Exponential Distribution

67. Example