Probabilities and Statistics

76 flashcards · Mathematics · by Athena Papkouf

  1. Given n distinct objects, the number of different permutations (without repetition) of length r ≤ n is

  2. permutations of length n

    n!

  4. multinomial coefficient

  5. binomial coefficient

  6. The number of ways of choosing a set of r objects from a set of n distinct objects without repetition is

  7. The number of ways of distributing n distinct objects into r distinct groups of size n1, . . . , nr, where n1 + · · · + nr = n, is

  8. Pascal’s triangle

  9. Vandermonde’s formula

  10. A probability space (Ω, F,P) is a mathematical object associated with a random experiment, comprising:

    a set Ω, the sample space (universe), which contains all the possible results (outcomes, elementary events) ω of the experiment; a collection F of subsets of Ω. These subsets are called events, and F is called the event space; a function P : F → [0, 1] called a probability distribution, which associates a probability P(A) ∈ [0, 1] to each A ∈ F

  11. An event space F is a set of the subsets of Ω such that:

  12. A probability distribution P assigns a probability to each element of the event space F, with the following properties:

  13. (7 items)

  14. P is called a continuous set function because

    for all sequences of sets for which

  15. Inclusion-exclusion formulae

  16. number of terms in the general inclusion-exclusion formula

  17. Let A, B be events of the probability space (Ω, F,P), such that P(B) > 0. Then the conditional probability of A given B is

    If P(B) = 0, we adopt the convention P(A ∩ B) = P(A | B)P(B), so both sides are equal to zero.

  18. Let (Ω, F,P) be a probability space, and let B ∈ F such that P(B) > 0 and Q(A) = P(A | B). Then (Ω, F, Q) is a probability space. In particular,

  19. Law of total probability

  20. Bayes theorem

  21. Prediction decomposition

    Let A1, . . . , An be events in a probability space. Then

  22. Let (Ω, F,P) be a probability space. Two events A, B ∈ F are independent (we write A ⊥⊥ B) if

  23. mutual independence

  24. pairwise independence

  25. conditional independence

  26. Does mutual independence entail pairwise independence?

    Mutual independence entails pairwise independence, but the converse is only true when n = 2

  27. Does mutual independence imply conditional independence?

    Mutual independence neither implies nor is implied by conditional independence.

  28. random variable

    Let (Ω, F,P) be a probability space. A random variable (rv) X : Ω → R is a function from the sample space Ω taking values in the real numbers R.

  29. support of random variable/ discrete random variable

  30. indicator variable/Bernoulli random variable/Bernoulli trial.

    A random variable that takes only the values 0 and 1

  31. probability mass function (PMF)

  32. binomial random variable

  33. geometric random variable

  34. Lack of memory

  35. Negative binomial distribution

  36. Hypergeometric distribution

  37. Discrete uniform distribution

  38. Poisson distribution

  39. Cumulative distribution function

  40. Transformations of discrete random variables

  41. Expectation (discrete)

  42. Expected value of a function

  43. variance

  44. conditional probability mass function

  45. Conditional expected value

  46. probability density function

  47. Uniform distribution

  48. exponential random variable

  49. expectation (continuous)

  50. variance (continuous)

  51. expectation of g(X)

  52. Conditional densities

  53. p quantile

  54. General transformation

  55. Normal distribution

  56. standard normal

  57. standard normalisation

  58. (joint) probability mass function

  59. (joint) cumulative distribution function

  60. (joint) density

  61. (joint) cumulative distribution function (continuous)

  62. marginal probability mass/density function

  63. conditional probability mass/density function

  64. Jjoint cumulative distribution function (multivariate)

  65. joint density/mass probability function

  66. multinomial distribution of denominator m and probabilities (p1, . . . , pk)

  67. Random variables X, Y defined on the same probability space are independent if

  68. expectation of g(X, Y )

  69. joint moments

  70. joint central moments

  71. covariance

  72. Properties of covariance

  73. correlation

    This measures the linear dependence between X and Y

  74. Let g(X, Y ) be a function of a random vector (X, Y ). Its conditional expectation given X = x is

  75. Expectation and conditioning (multiple variables)