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Statistics for Machine Learning > 1. Fundamentals of Probability > Flashcards

1. Fundamentals of Probability Flashcards

Fundamentals of probability for Machine Learning

1
Q

What is probability?

A

Probability is a measure of the likelihood that an event will occur, ranging from 0 to 1.

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2
Q

What is a sample space?

A

The set of all possible outcomes of a probabilistic experiment.

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3
Q

What is an event in probability?

A

A subset of the sample space, representing one or more outcomes.

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4
Q

What are the three axioms of probability?

A
  1. P(A) ≥ 0 for any event A.
  2. P(S) = 1 where S is the sample space.
  3. P(A ∪ B) = P(A) + P(B) for disjoint events A and B.
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5
Q

What is conditional probability?

A

The probability of event A given that event B has occurred, denoted P(A|B).

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6
Q

What is the formula for conditional probability?

A

P(A|B) = P(A ∩ B) / P(B)
provided P(B) > 0.

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7
Q

What is the law of total probability?

A

P(A) = Σ P(A|B_i)P(B_i) for a partition {B_i} of the sample space.

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8
Q

What is Bayes’ Theorem?

A

P(A|B) = [P(B|A) * P(A)] / P(B).

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9
Q

What is an example of Bayes’ Theorem in real life?

A

Medical testing: Given a positive test result, what is the probability of having the disease?

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10
Q

What does it mean for two events to be independent?

A

Two events A and B are independent if P(A ∩ B) = P(A)P(B).

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11
Q

What is the difference between independent and dependent events?

A

Independent events do not affect each other’s probability, whereas dependent events do.

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12
Q

How do you check if events A and B are independent?

A

Check if P(A|B) = P(A) or if P(A ∩ B) = P(A)P(B).

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13
Q

What is a random variable?

A

A function that assigns numerical values to outcomes of a probabilistic experiment.

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14
Q

What is the difference between discrete and continuous random variables?

A

Discrete variables take countable values, while continuous variables take any value in an interval.

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15
Q

What is the expected value (mean) of a random variable?

A

E[X] = Σ x * P(X=x) for discrete variables or ∫ x f(x) dx for continuous variables.

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16
Q

What is a probability mass function (PMF)?

A

A function that gives the probability of each possible value for a discrete random variable.

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17
Q

What is a probability density function (PDF)?

A

A function describing the likelihood of a continuous random variable taking a value.

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18
Q

What is a cumulative distribution function (CDF)?

A

A function that gives the probability that a random variable is less than or equal to a given value.

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19
Q

Define probability.

A

A measure of the likelihood of an event occurring, between 0 and 1.

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20
Q

What is a trial in probability?

A

A single performance or observation of an experiment.

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21
Q

What is an outcome in probability?

A

A single possible result from a probability experiment.

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22
Q

What is an experiment in probability?

A

A process that leads to one of several possible outcomes.

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23
Q

What is the sample space of rolling a fair six-sided die?

A

{1, 2, 3, 4, 5, 6}.

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24
Q

What is an event in probability?

A

A subset of the sample space.

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25
Q

What are mutually exclusive events?

A

Events that cannot occur simultaneously (P(A ∩ B) = 0).

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26
Q

What is an example of mutually exclusive events?

A

Getting heads and tails on a single coin flip.

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27
Q

What is a certain event?

A

An event that has a probability of 1 (P(A) = 1).

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28
Q

What is an impossible event?

A

An event that has a probability of 0 (P(A) = 0).

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29
Q

What is the complement rule?

A

P(A’) = 1 - P(A), where A’ is the complement of A.

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30
Q

What is an exhaustive set of events?

A

A set of events that covers the entire sample space.

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31
Q

What is conditional probability?

A

The probability of an event occurring given another event has occurred.

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32
Q

How is conditional probability written?

A

P(A|B), meaning the probability of A given B.

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33
Q

State the formula for conditional probability.

A

P(A|B) = P(A ∩ B) / P(B), if P(B) > 0.

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34
Q

What does it mean if P(A|B) = P(A)?

A

It means that events A and B are independent.

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35
Q

What does the law of total probability state?

A

P(A) = Σ P(A|B_i)P(B_i) over a partition {B_i}.

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36
Q

How does conditional probability relate to joint probability?

A

P(A ∩ B) = P(A|B)P(B).

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37
Q

State Bayes’ Theorem.

A

P(A|B) = [P(B|A) * P(A)] / P(B).

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38
Q

What does Bayes’ Theorem help with?

A

Updating probabilities based on new evidence.

39
Q

Give an example of Bayes’ Theorem in machine learning.

A

Used in Naive Bayes classifiers for text classification.

40
Q

What is prior probability?

A

The initial probability of an event before new evidence is introduced.

41
Q

What is likelihood in Bayes’ Theorem?

A

P(B|A), the probability of B occurring given A.

42
Q

What is the posterior probability?

A

The updated probability of A after considering B.

43
Q

What are independent events?

A

Events where P(A ∩ B) = P(A)P(B).

44
Q

What is the multiplication rule for independent events?

A

P(A ∩ B) = P(A)P(B).

45
Q

How can you check if two events are independent?

A

If P(A|B) = P(A), then A and B are independent.

46
Q

What are dependent events?

A

Events where the occurrence of one affects the probability of the other.

47
Q

Give an example of dependent events.

A

Drawing two cards from a deck without replacement.

48
Q

Define a random variable.

A

A function that assigns numerical values to outcomes of an experiment.

49
Q

What is a discrete random variable?

A

A variable that takes a countable number of values.

50
Q

What is a continuous random variable?

A

A variable that takes infinitely many values in an interval.

51
Q

What is the expectation (mean) of a discrete random variable?

A

E[X] = Σ x * P(X=x).

52
Q

What is variance?

A

Var(X) = E[(X - E[X])²], a measure of spread of a random variable.

53
Q

What is the standard deviation?

A

The square root of variance, a measure of dispersion.

54
Q

What is a probability mass function (PMF)?

A

A function giving the probability of each value for a discrete random variable.

55
Q

What is a probability density function (PDF)?

A

A function giving the relative likelihood of a continuous random variable taking a value.

56
Q

What is a cumulative distribution function (CDF)?

A

A function giving P(X ≤ x) for a random variable X.

57
Q

What is the relationship between a CDF and a PDF?

A

The derivative of a CDF is the PDF.

58
Q

What is the expected value of a probability distribution?

A

A weighted average of possible values, using probabilities as weights.

59
Q

What is the law of large numbers?

A

As the number of trials increases, the sample mean converges to the expected value.

60
Q

What is the central limit theorem?

A

For large n, the sample mean distribution approaches a normal distribution.

61
Q

What is the difference between a population and a sample?

A

A population is the entire set of elements, while a sample is a subset.

62
Q

What is the difference between a parameter and a statistic?

A

A parameter describes a population; a statistic describes a sample.

63
Q

What is a probability generating function?

A

A function that generates the probabilities of a discrete random variable.

64
Q

What is a moment generating function?

A

A function that generates the moments (mean, variance, etc.) of a probability distribution.

65
Q

What is a probability space?

A

A mathematical triplet (Ω, F, P) where Ω is the sample space, F is the event space, and P is the probability function.

66
Q

What is a discrete probability distribution?

A

A probability distribution that deals with countable outcomes.

67
Q

What is a continuous probability distribution?

A

A probability distribution that deals with outcomes over a continuous range.

68
Q

What is the probability of an event A not occurring?

A

P(A’) = 1 - P(A).

69
Q

What is the sum of probabilities of all possible outcomes?

A

Always equals 1.

70
Q

What is the difference between theoretical and empirical probability?

A

Theoretical is based on expected outcomes, while empirical is based on actual experiments.

71
Q

What is the principle of equally likely outcomes?

A

If all outcomes are equally likely, then P(A) = |A|/|S|.

72
Q

How does probability relate to frequency?

A

Probability approximates frequency in a large number of trials.

73
Q

What does P(A|B) = 0 mean?

A

Event A never occurs given that B has occurred.

74
Q

What does P(A|B) = 1 mean?

A

Event A always occurs given that B has occurred.

75
Q

What does it mean if P(A|B) > P(A)?

A

B increases the likelihood of A.

76
Q

What does it mean if P(A|B) < P(A)?

A

B decreases the likelihood of A.

77
Q

What is an application of Bayes’ Theorem in medicine?

A

Used to calculate the probability of having a disease given a positive test result.

78
Q

What is the naive Bayes classifier?

A

A machine learning algorithm that assumes independence between features.

79
Q

What is the support of a probability distribution?

A

The set of all values where the probability function is nonzero.

80
Q

What is Chebyshev’s inequality?

A

It states that at least (1 - 1/k²) of values lie within k standard deviations of the mean.

81
Q

What is the moment of a distribution?

A

A quantitative measure of the shape of a distribution.

82
Q

What is the probability of a uniform distribution over [a, b]?

A

1 / (b - a).

83
Q

What is the mode of a distribution?

A

The value that appears most frequently.

84
Q

What is the median of a distribution?

A

The value that splits the probability distribution into two equal halves.

85
Q

What is the main consequence of the CLT?

A

The sample mean approximates a normal distribution as the sample size grows.

86
Q

Why is CLT important in statistics?

A

It justifies the use of normal approximations in hypothesis testing and confidence intervals.

87
Q

What is Jensen’s inequality?

A

For a convex function g, E[g(X)] ≥ g(E[X]).

88
Q

What is the covariance of X and Y?

A

E[(X - E[X])(Y - E[Y])].

89
Q

What is correlation?

A

A normalized measure of covariance, ranging from -1 to 1.

90
Q

What does a correlation of 0 mean?

A

No linear relationship between X and Y.

91
Q

What is entropy?

A

A measure of uncertainty in a probability distribution.

92
Q

What is Kullback-Leibler (KL) divergence?

A

A measure of how one probability distribution diverges from another.

93
Q

What is the relationship between entropy and information gain?

A

Information gain is the reduction in entropy after observing a variable.

Statistics for Machine Learning (9 decks)

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