Chapter 3 Discrete Random Variables

1. Chapter 3Discrete Random Variables 主講人:虞台文

2. Content • Random Variables • The Probability Mass Functions • Distribution Functions • Bernoulli Trials • Bernoulli Distributions • Binomial Distributions • Geometric Distributions • Negative Binomial Distributions • Poisson Distributions • Hypergeometric Distributions • Discrete Uniform Distributions

3. Chapter 3Discrete Random Variables Random Variables

4. Definition  Random Variables A random variableX of a probability space (, A, P) is a real-valued function defined on , i.e.,

5. 原來熊貓不是貓 Definition  Random Variables A random variableX of a probability space (, A, P) is a real-valued function defined on , i.e.,

6. Example 1

7. 1 Example 1

8. 2 Example 1

9. Example 1

10. l a b Example 2

11. l a b Example 2

12. Notations

13. Example 1

14. Example 1

15. Example 1

16. Definition Discrete Random Variables A discrete random variableX is a random variable with range being a finite or countable infinite subset {x1, x2, . . .} of real numbers R.

17. countable uncountable What is countablity? Definition Discrete Random Variables A discrete random variableX is a random variable with range being a finite or countably infinite subset {x1, x2, . . .} of real numbers R. The set of all integers The set of all real numbers

18. Example 1 All are finite X, Y and Z are discrete random variables.

19. l a b Example 2 All are uncountable X, Y and Z are notdiscrete random variables.

20. l a b Example 2 In fact, they are continuous random variables. All are uncountable X, Y and Z are notdiscrete random variables.

21. Chapter 3Discrete Random Variables The Probability Mass Functions

22. Definition The Probability Mass Function (pmf) The probability mass function (pmf) of r.v. X, denoted by pX(x), is defined as

23. Example 4

24. x 2 3 4 5 6 7 8 9 10 11 12 6/36 5/36 4/36 3/36 2/36 1/36 Example 4

25. y 1 2 3 4 5 6 Example 4

26. z 0 1 Example 4

27. Properties of pmf’s • x

28. Chapter 3Discrete Random Variables Distribution Functions

29. Cumulative Distribution Function (cdf) cdf

30. Cumulative Distribution Function (cdf) pmf cdf

31. pX(x) x x1 x2 x3 x4 FX(x) x x1 x2 x3 x4 pmf cdf Cumulative Distribution Function (cdf)

32. FX(x) x x1 x2 x3 x4 pmf cdf Cumulative Distribution Function (cdf) pX(x) x x1 x2 x3 x4

33. FX(x) x x1 x2 x3 x4 pmf cdf Cumulative Distribution Function (cdf) pX(x) x x1 x2 x3 x4

34. FX(x) x x1 x2 x3 x4 pmf cdf Cumulative Distribution Function (cdf) pX(x) x x1 x2 x3 x4 1 pX(x4) pX(x3) pX(x2) pX(x1)

35. Example 5

36. x 2 3 4 5 6 7 8 9 10 11 12 6/36 5/36 4/36 3/36 2/36 1/36 Example 5

37. p(x) x F(x) x Example 5

38. p(x) x F(x) x Example 5

39. y 1 2 3 4 5 6 Example 5

40. p(y) y F(y) y Example 5

41. Properties of cdf’s • x • Monotonically nondecreasing.

42. Properties of cdf’s • x • Monotonically nondecreasing. F(b) F(a)

43. Chapter 3Discrete Random Variables Bernoulli Trials

44. Bernoulli Trials • Suppose an experiment consists of n trials, n> 0. • The trials are called Bernoulli trials if three conditions are satisfied: • Each trial has a sample space {S=1, F=0} (two outcomes), Sto be called success and F to be called failure. • For each trial P(S) = p and P(F) = q, where 0 ≤ p ≤ 1 and q = 1 − p. • The trials are independent.

45. Is this experiment to performing Bernoulli Trials? Why? Example 6 Tossing a die ten times, the actual face number in each toss is unnoted. Instead, the outcome of 1 or 2 will be considered a success, and the outcome of 3, 4, 5, or 6 will be considered a failure. What is the sample space of the experiment?

46. Discussion What probabilities may interest us on performing Bernoulli Trials?

47. Chapter 3Discrete Random Variables Bernoulli Distributions

48. Bernoulli Distributions • Let r.v. X denote the outcome of a Bernoulli trial, and let the probability of success equal to p. • Then, we have pmf cdf

49. Bernoulli Distributions • Let r.v. X denote the outcome of a Bernoulli trial, and let the probability of success equal to p. • Then, we have pmf cdf

50. pX(x) 1p p x 0 1 Bernoulli Distributions • Let r.v. X denote the outcome of a Bernoulli trial, and let the probability of success equal to p. • Then, we have pmf cdf