Introduction to Probability, Statistics, and Random Processes (Hossein Pishro-Nik) Chapter 1

61 flashcards · Mathematics · by charlie42

1. What's a set?

   A collection of things (elements)

2. How to read |?

   "Such that"

3. Alternative notation for |

   :

4. Universal set for throwing a six-side die

   S = {1, 2, 3, 4, 5, 6}

5. Notation for union of n sets

6. The union of two sets A and B is a set containing...

   ...all elements that are in A or in B

7. What's this: A∩B ?

   An intersection of A and B

8. What does the shaded area shows?

   Complement of A

9. The set A−B consists of elements that are...

   in A but not in B

10. Visualize that A−B = A∩Bc

11. Two sets that do not have any shared elements (2)

    Mutually exclusive Disjoint A∩B = ∅

12. A collection of non-empty sets A1, A2,... is a partition of a set A if they are:

    disjoint, and their union is A

13. De Morgan's law

14. De Morgan's law with 2 sets, visualize

15. Distributive law (sets)

16. A Cartesian product of two sets A and B is written as...

    A×B

17. A Cartesian product of two sets A and B is the set...

    containing ordered pairs from A and B

18. Cartesian product in set-builder notation

19. Number of elements notation

    |A|

20. The multiplication principle (2 sets)

    |A×B|=MN M - number of elements in A N - number of elements in B

21. Cartesian product of n sets A1,A2,⋯,An

22. The multiplication principle for finite sets

23. Cardinality of a set

    Size of a set

24. Find |A| if A = {1,2,3}

    3

25. Inclusion-exclusion principle for two sets

26. Inclusion-exclusion principle for three sets. Visualize

27. 2 types of infinite sets and examples

    countable (ℕ, ℤ, ℚ, and any of their subsets) uncountable (any set containing an interval on the real line such as [a,b],(a,b],[a,b), or (a,b), where a<b)

28. Set A is called countable if one of the following is true:

    if it is a finite set, ∣A∣<∞; or it can be put in one-to-one correspondence with natural numbers ℕ, in which case the set is said to be countably infinite. A set is called uncountable if it is not countable.

29. A function f is...

    a rule that takes an input from a specific set (called the domain), and produces an output from another set (called co-domain).

30. A function maps elements from the domain set to elements in the co-domain with the property that...

    each input is mapped to exactly one output.

31. If A is the domain and B is the co-domain for the function f, we use the following notation:

32. Domain and co-domain of f(x)=x2

33. The output of a function f:A→B always belongs to...

    the co-domain B

34. What's the range of a function?

    The range of a function is the set containing all the possible values of f(x).

35. Function range notation

    Range(f)

36. Range of f(x)=x2 in set notation

37. Visualize range, domain, and co-domain of a function

38. Let S={1,2,3}. Write all the possible partitions of S

39. Find the range of the function f:ℝ→ℝ defined as f(x)=sin(x).

    Range(f)=[−1,1]

40. Definition of a random experiment

    Random experiment is a process by which we observe something uncertain

41. After a random experiment, the result of the experiment is...

    known

42. Outcome definition

    Outcome is a result of a random experiment

43. Sample space definition

    Set of all possible outcomes of a random experiment

44. Sample space of this random experiment: observe the number of iPhones sold by an Apple store in Boston in 2015

    S={0,1,2,3,⋯}

45. Trial definition

    A trial is a particular performance of a random experiment

46. Event definition, example

    A subset of the sample space E={2,4,6} (outcome of rolling a die is an even number)

47. If A and B are events, then ... and ... are also events.

    A∪B, A∩B

48. Event A∪B occurs if

    A or B occur

49. Event A∩B occurs if

    both A and B occur

50. Similarly, if A1,A2,⋯,An are events, then the event A1∪A2∪A3⋯∪An occurs if

    at least one of A1,A2,⋯,An occurs.

51. The event A1∩A2∩A3⋯∩An occurs if

    all of A1,A2,⋯,An occur.

52. The key words "or" and "at least" correspond to

    unions

53. The key words "and" and "all of" correspond to

    intersections

54. We assign a probability measure P(A) to an event A. This is a...

    value between 0 and 1 that shows how likely the event is

55. How many axioms of probability are there

    3

56. Axiom of probability 1

    For any event A, P(A) ≥ 0

57. Axiom of probability 2

    Probability of the sample space S is P(S)=1.

58. Axiom of probability 3

    If A1,A2,A3,⋯ are disjoint events, then P(A1∪A2∪A3⋯)=P(A1)+P(A2)+P(A3)+⋯

59. Example: In a presidential election, there are four candidates. Call them A, B, C, and D. Based on our polling analysis, we estimate that A has a 20 percent chance of winning the election, while B has a 40 percent chance of winning. What is the probability that A or B win the election?

    Events that {A wins}, {B wins}, {C wins}, and {D wins} are disjoint (more than one of them cannot occur at the same time). From the third axiom of probability, the probability of the union of two disjoint events is the summation of individual probabilities:

60. 3 ways to write the probability of intersection of A and B

    P(A∩B) = P(A and B) = P(A,B) = P(AB)

61. Two ways to read this: P(A∪B)

    Probability of A or B Probability of union of A and B