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Statistics & Probability > Statistics & Probability > Flashcards

Statistics & Probability Flashcards

1
Q

Element

A

Possible outcome of a random experiment

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2
Q

Sample space

A

A set containing all possible outcomes

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3
Q

Event

A

Set of outcomes of a random experiment

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4
Q

Probability of an event (definition)

A

A measurement of the likehood of observing the event

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5
Q

Mutually exclusive events

A

If an event A occurs, then an event B cannot occur

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6
Q

Independent events

A

Events that occur independently of each other

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7
Q

Singleton event

A

An event with only 1 outcome

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8
Q

Probability (formula)

A

P(A) = |A| / |Ω|,

where:
|A| = size of an event
|Ω| = size of a sample space

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9
Q

Multiplication principle of counting

A

If there are N1 ways of performing task 1 and N2 ways of performing task 2, then there are N1*N2 ways of performing both tasks

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10
Q

Additive principle (for disjoint events)

A

If event A can occur in m ways and an event B can occur in n ways, than the event “A or B” can occur in m+n ways

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11
Q

Permutation (definition)

A

Possible rearrangement of objects

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12
Q

Permutation (formula)

A

n!

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13
Q

K permutations of n elements (definition)

A

Number of ways to arrange k objects picked from n distinct objects

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14
Q

nPk formula

A

nPk = n! / (n-k)!

Order matters

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15
Q

0! = …

A

1

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16
Q

nCk formula

A

nCk = n! / [(n-k)!k!]

Order is not important

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17
Q

Complement of an event

A

An event of not observing the included outcomes

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18
Q

(A ∩ B)c

A

Ac ∪ Bc

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19
Q

P(A∩B) for mutually exclusive events

A

0

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20
Q

P(A∩B) for independent events

A

P(A)*P(B)

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21
Q

P(A∪B)

A

P(A)+P(B) - P(A∩B)

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22
Q

∩

A

intersection (and)

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23
Q

∪

A

union (or)

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24
Q

Conditional probability (definition)

A

A way of calculating probability of an event using prior information

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25
Q

Conditional probability (formula)

A

P(A|B) = P(A∩B) / P(B)

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26
Q

Law of total probability (definition)

A

If the probability of an event is unknown it can be calculated using the known probabilities of several distinct events

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27
Q

Law of total probability (formula)

A

P(A) = ∑n P(A∩Bn) = ∑ P(A|Bn)*P(Bn)

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28
Q

Bayes theorem (definition)

A

The probability of an event given prior knowledge of related events that occurred earlier

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29
Q

Bayes theorem (formula)

A

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

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30
Q

Derangement (definition)

A

Permutation of a set where none of the elements appear in the original positions

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31
Q

Derangement (formula)

A

!n = n!∑n [-i^i] / [i!]
i = 0

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32
Q

Random variable

A

Variable whose value is determined by a random experiment

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33
Q

Discrete probability distribution

A

Table or formula that lists the probabilities for each outcome of the random variable X

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34
Q

Total area under the curve in normal distribution = …

A

1

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35
Q

The role of standard deviation in normal distribution

A

Std. dev describes the width of the curve: wider the std dev is, the wider curve is

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36
Q

dy/dx CDF = …

A

PDF

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37
Q

.. ∫PDF = …

A

CDF

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38
Q

Binomial distribution (definition)

A

Statistical distribution that represents the probability for X success in N trials, given a success probability P for each trial

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39
Q

Binomial distribution (formula)

A

P(X) = nCx * p ^x * q^(n-x)

x = number of success events
n = number of trials
p = probability of success
q = probability of failure (e.g. 1-p)

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40
Q

Variance of binomial distribution

A

np(1-p)

n= number of trials
p = probability of success

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41
Q

Expected value of binomial distribution

A

n*p

n= number of trials
p = probability of success

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42
Q

P(A∩B) for dependent events

A

P(B|A) * P(A)

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43
Q

Bernoulli trial

A

An experiment whose outcome is random and can be either of 2 possibilities (success or failure)

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44
Q

“At least” probability

A

All probabilities LARGER than the given probability

45
Q

“At most” probability

A

All probabilities SMALLER than the given probability (incl. 0)

46
Q

Poisson distribution (definition)

A

Probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently

47
Q

Poisson distribution if a time interval is different

A

mean2/mean1 = time2/time1

48
Q

Normal distribution (definition)

A

A probability distribution that is symmetric about the mean, showing that data near the mean are more frequent to occur than data far from the mean

49
Q

Z-score (definition)

A

Value that tells how many standard deviations a number is above or below the mean

50
Q

Z-score (formula)

A

z = [X - mean] / st.dev

51
Q

Standard normal distribution

A

Normal distribution with a mean=0 and a standard deviation=1

52
Q

Standard deviation (definition)

A

A measure of how far a group of numbers is from the mean

53
Q

Variance (definition)

A

The average of the squared differences from the mean

54
Q

Probability mass function

A

A function that gives the probability that a discrete random variable will be exactly equal to specific value

55
Q

Probability density function

A

A function that provides the likehood that the value of a continuous random variable will fail between a certain range of values

56
Q

Cumulative distribution function

A

P(x<=x)

A probability that X takes a value less than or equal to x

57
Q

Mean

A

The sum of all values of a collection of data divided by the total number of values in the data

58
Q

Geometric distribution (definition)

A

A discrete probability distribution that represents the probability of successive failures before a first success is obtained in a Bernoulli trial

59
Q

Geometric distribution (formula)

A

P(X=x) = p*q^[x-1]

p = probability of success
q = probability of failure
x = number of trials before 1st success

60
Q

Geometric distribution (CDF)

A

P(X<=x) = 1-q^x

61
Q

Mean for geometric distribution

A

1/p

62
Q

Exponential distribution

A

Probability distribution of time between events in the Poisson process; amount of time until some specific event happens

63
Q

Memoryless property

A

A given probability distribution is independent of its history. Any time may be marked as time zero

64
Q

Poisson process features

A
  1. Average time between events is known and constant
  2. Events are independent of each other
  3. Events cannot occur simultaneously
65
Q

Binomial experiment

A

An experiment consisting of a fixed number of independent Bernoulli trials

66
Q

Negative binomial distribution (definition)

A

A distribution of the number of trials needed to get Rth success;

Goal: find the specific success event in combination with previous needed successes

67
Q

Discrete random variable

A

A random variable which takes only finite or countably infinite values

68
Q

Bernoulli distribution

A

A discrete probability distribution that models a random variable with only 2 possible outcomes

69
Q

Continuous uniform distribution

A

A probability distribution that describes an experiment in which the outcomes of the random variable have equally likely probabilities of occurring within an interval [a,b]

70
Q

PDF for continuous uniform distribution

A

1 / (b-a) when x [a,b]

71
Q

CDF for an exponential distribution

A

1 - [e^(-λx)]

72
Q

Gamma distribution

A

Continuous distribution that is used to estimate the precision of a normal distribution by incorporating prior knowledge

73
Q

Student T distribution

A

Continuous distribution that is used for estimating population parameters for small sample sizes or unknown variances

74
Q

Arrange n people in a circle

A

(n-1)!

75
Q

Standard deviation for binomial distribution (formula)

A

sqrt(npq)

76
Q

Hypergeometric distribution (definition)

A

A discrete probability distribution that is used to calculate probability of obtaining a certain number of successes from a finite population where every draw is done without replacement

77
Q

Characteristics of a hypergeometric experiment

A

1) Take a samples from 2 groups
2) Concerned with a group of interest (e.g. 1 st group)
3) Sampling without replacement
4) Not dealing with Bernoulli trials

78
Q

Hypergeometric distribution (formula)

A

P(X=k) = [KCk*(N-K)C(n-k)] / [NCn]

where
K = number of successes in population
k = number of observed successes
N = population size
n = number of draws

79
Q

Collectively exhaustive events

A

Events cover all possible outcomes and one of the events in the set must occur

80
Q

Joint probability

A

Statistical measure that calculates the likehood of 2 events occurring together and the same point in time

81
Q

Joint PMF

A

A function of several variables that gives the probability of values of tuples of random variables

82
Q

Joint PDF

A

A function of several variables that gives the density of values of tuples of random variables

83
Q

Joint probability measure

A

A probability measure for a tuple (pair, triple etc…) of random variables

84
Q

Joint CDF

A

A function of several variables that gives the accumulated probability across a range of values for tuples of random variables

85
Q

Compound probability

A

The likehood of two independent events occurring (P(A)*P(B))

86
Q

Marginal distribution (for X)

A

The sum of the joint distribution over all the possible values of Y

87
Q

Independence of two discrete random variables

A

Two discrete random variables are independent if their joint PMF is the same as the product of each marginal PMF

88
Q

Bivariate uniform distribution PDF

A

1 / [(b-a)(d-c)] if (x,y) in [a,b][c,d]

89
Q

Conditional PMF Px|y (x|y)

A

P({X=x} ∩ {Y=y}) / Py(Y=y)

90
Q

Expected value

A

a long term average of a random variable based on the values it takes and its corresponding probabilities

91
Q

Expected value for discrete r.v.

A

E[X] = ∑Xi*f(Xi)

92
Q

Expected value for continuous r.v.

A

E[X] = ∫X*f(X)dx

93
Q

Properties of expectations

A

1) The expected value of a constant C = C
2) Linearity: E[aX+b] = aE[X]+b

94
Q

E[aX+bY]

A

aE[X] + bE[Y]

95
Q

E[X+Y]

A

E[X]+E[Y]

96
Q

Expectation of independent random variables

A

E[XY] = E[X]E[Y]

97
Q

Conditional expectation

A

The expected value of X conditioned to the event Y=y describes the long-term event conditioned to the event that Y may take the value y

98
Q

Sample variance

A

A measure of the variation of the values comparing each data point to the mean

99
Q

Variance (formula)

A

Var[X] = E[X^2] - [E[X]^2]

100
Q

The variance of constant

A

0

101
Q

Var[aX]

A

a^2 Var[X]

102
Q

Var[aX+c]

A

a^2 Var[X]

103
Q

Var[X+Y]

A

Var[X]+Var[Y]

104
Q

Covariance

A

The measurement of the directional relationship between two random variables

105
Q

Covariance for independent random variables

A

0

106
Q

Covariance formula

A

Cov[X,Y] = E[XY] - E[X]E[Y]

107
Q

Central moments

A

1 = Mean
2 = Variance
3 = Skewness
4 = Kurtosis

108
Q

Moment generation function

A

A random variable which gives an alternative route for the computation of expectation values, variances, probability measures etc.

Statistics & Probability (1 decks)

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