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Prelims > Probability And Statistics Equations > Flashcards

Probability And Statistics Equations Flashcards

1
Q

Axioms of Probability

A
  1. P(A)>=0 for all A is a member of F
  2. P(omega)=1
  3. If A1, A2, … are members of F and are mutually exclusive P(A1 U A2 U…) = P(A1) + P(A2) + …
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2
Q

Complement rule

A

P(Ac)= 1- P(A)

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3
Q

Probability of the Union of Two Events Rule

A

P(AUB) = P(A) + P(B) - P(AmetszetB)

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4
Q

Bounds on Probabilities Rule

A

P(AUB) =< P(A) + P(B)

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5
Q

Logical Consequence Rule

A

If B logically entails A, then P(A)>= P(B)

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6
Q

Conditional Probability

A

P(AIB) = P(AmetszetB)/P(B)

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7
Q

Axioms revised based on conditional probability

A
  1. 0=< P(AIB) =< 1
  2. P(BIB) = 1
  3. If A1, A2, … are mutually exclusive given B, then P(A1, A2, … IB) = P(A1IB) + P(A2IB)…
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8
Q

Law of Total Probability

A

P(A) = P(AmetszetB1) + P(AmetszetB2) = P(AIB1)P(B) + P(AIB2)P(B)

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9
Q

Two events are independent if

A

P(AmetszetB) = P(A)P(B) OR P(AIB)= P(A)

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10
Q

Bayes’ Rule

A

P(AIB) = P(BIA)P(A)/P(B)

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11
Q

PMF definition

A
  1. f(x) = P(X=x)
  2. f(x) >= 0
  3. SUMf(x)=1
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12
Q

PMF of N independent Bermoulli (coin toss) variables with parameter p

A

f(x) = Kp^x(1-p)^N-x
K is a scaling factor independent of N or p

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13
Q

PDF definition

A

f(x) = lim(h->0) P(x=< X =< x+h)/h
1. f(x) >= 0
2. Integral fx dx = 1

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14
Q

Uniform distribution U(a,b) PDF

A

If a=<x=<b f(x) = 1/b-a
Otherwise f(x) = 0

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15
Q

Normal Distribution PDF

A

f(x) = 1/sigma*gyokalatt2pi exp(-1/2 (x-nu)^2/sigma^2)

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16
Q

CDF Definition

A

F(x) = P(X=<x) for all x in support of X
1. 0=<F(x)=<1
2. F(x) is a non-decreasing function in x
3.P(X>x) = 1 - P(X=<x)
4. P(a<x=<b) = F(b) - F(a)

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17
Q

Uniform Distribution CDF

A

If x<a F(x)= 0
If x is a member of Ia,bI F(x)=x-a/b-a
If b<x F(x)=1

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18
Q

Connection of CFD and PDF

A

f(x)= dF(x)/dx

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19
Q

Standardization N(0,1) Standard Normal Distribution

A

z=(X-nu)/sigma
P(X=<z) in SN Table

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20
Q

Joint CDF

A

F(x,y) = P(X=<x, Y=<y)

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21
Q

Joint PDF/PMF

A

Discrete f(x,y) = P(X=x, Y=y)
Continous f(x, y) = d^2F(x,y)/dxdy

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22
Q

Marginal PDF (of x)

A

f(x) = f(x, whatever the value of y)

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23
Q

If X and Y are independent (relationship between joint PDF and product of marginals)

A

f(x,y) = f(x)g(x) - joint PDF = product of marginals
F(x,y)= F(x)G(y) - joint CDF = product of marginals

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24
Q

Conditional PDF, CDF

A

Slices (impact of variation in one variable on the probability of another)

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25
Q

Expected value E[X]

A

Discrete E[X]=SUMi xif(xi)
Continous E[X]=integralxf(x)dx

Probability weighted sums of possible values of x

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26
Q

Uniform Expected value - U(a,b)

A

E[X]=(a+b)/2

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27
Q

Normal - Expected Value N(nu, sigma^2)

A

E[X]=nu

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28
Q

Expected value of a function of a variable

A

Discrete E[g(X)] = SUMi g(xi)f(xi)
Continuous E[g(X)] = integral g(x)f(x)dx

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29
Q

Affine functions (E[a+bX])

A

E[a+bX]=a+bE[X]

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30
Q

Addition of expectations g(x) and h(y) is a function of another or the same variable

A

E[g(x)+h(y)]=E[g(x)] + E[h(y)]

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31
Q

Multiplication of expectations IF INDEPENDENT

A

E[g(x)h(y)]= E[g(x)]E[h(y)]

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32
Q

Jensen’s inequality

A

If f is concave E[f(X)] =< f(E[X])
If f is convex E[f(X)] >= f(E[X])

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33
Q

Variance

A

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

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34
Q

Variance with nu (expected value of the variable)

A

X: discrete Var(X) = SUMi(xi-nu)^2f(x)
X: continous Var(X) = integral (X - nu)^2f(x) dx

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35
Q

Standard Deviation

A

SD = gyok alatt Var(X)

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36
Q

Variance of Uniform

A

Var(X) = (b-a)^2/12

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37
Q

Variance of Normal

A

Var(X) = sigma^2

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38
Q

Variance of Binary Variable

A

Var(X)=p(1-p)

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39
Q

Conditional Expectation

A

Y discrete E(YIX=x) = SUMi yi f(yiIX)
Y continous E(YIX=x) = integral yf(yIx)dy

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40
Q

If E[h(X)YIX] =
Conditioning on X —> same as if it was a constant (linear transformation)

A

E[h(X)YIX] = h(X)E[YIX]

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41
Q

E[YIX] =
If X and Y are independent

A

E[YIX] = E[Y]

42
Q

Law of Iterated Expectations

A

E[Y] = E(E[YIX])

43
Q

Covariance

A

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

44
Q

Cov(X,X)

A

Cov(X,X)=Var(X)

45
Q

Cov(X,Y)

A

Cov(Y,X)

46
Q

Cov(X,a)

A

0

47
Q

Cov(aX,Y)

A

aCov(X, Y)

48
Q

Cov(X,Y+Z)=

A

Cov(X, Y) + Cov(X, Z)

49
Q

Var(aX+-bY)

A

a^2Var(X) + b^2Var(Y) +- 2abCov(X,Y)

50
Q

If X and Y are independent Cov(X, Y)= E[XY] - E[X]E[Y] =

A

0
E[XY] = E[X]E[Y]

51
Q

Correlation

A

Corr(X, Y) =
Cov(X, Y)/gyokVar(X)gyokVar(y) =
Cov((x-E[X])/gyokVar(X), (y-E[Y])/gyokVar(Y))

52
Q

Population

A

Complete enumeration of a same set of interest

53
Q

Sample

A

Subset of a population

54
Q

Sample frame

A

Source material or device from which a sample is drawn

55
Q

Simple Random Sampling (SRS)

A

Selects the pre-determined number of respondents to be interviewed from a target population with each potential respondent having an equal non-zero chance of being selected

56
Q

“Representative” sample

A

If the sampling procedure is repeated many times, the features of the sample would on average (across all the samples) match those of the population

57
Q

Quota sampling

A

Fixed quotas of certain types of respondents to be interviewed such that the resulting sample characteristics resemble those of the population

58
Q

Random variable

A

Deterministic functions which assign numbers to uncertain events which are generated by random sampling

59
Q

Independently and Identically Distributed (iid)

A

When sample values are all drawn from the same population and have the same distribution

60
Q

Parameter

A

Numerical measure that describes a specific characteristic of a population

61
Q

Statistic

A

Numerical measure that describes a specific characteristic of a sample. Formally a statistic is a function of a random variable (subject to sampling variation)

62
Q

Estimand

A

Parameter in the population which is to be estimated in a statistical analysis

63
Q

Estimator

A

A function for calculating an estimate of a given population parameter based on randomly sampled data. An estimator is a function of a sample of data which is drawn randomly. Different random samples result in different values for the estimator. They are themselves random variables and therefore have distributions, expected values etc.

64
Q

Estimate

A

An estimate is the numerical value of the estimator given a specific sample is draen; it is a nonrandom number (eg. The sample mean)

65
Q

Sampling distribution

A

Distribution of the estimator

66
Q

Standard error

A

A measure of variation in the sampling distribution; it is equal to the square root of the variance of the statistic

67
Q

Standard Deviation

A

A measure of variation in data it is equal to the square root of the variance of the data

68
Q

Law of Large Numbers

A

As the sample size grows the sample mean converges, in a certain sense, to the population mean

69
Q

Central Limit Theorem

A

As the sample size grows the sampling distribution of the standardised sample mean converges to a standard normal N(0,1)

70
Q

Nuisance parameter

A

Any parameter which is not of immediate interest but which must be accounted for in the analysis of those parameters which are of interest

71
Q

General form of confidence interval

A

“Sample statistic +- a number of std errors * std error of the statistic”

72
Q

Standard error for 0.9 probability

A

+-1.645

73
Q

Number of standard errors for probability 0.95

A

+- 1.96

74
Q

Number of standard errors for probability 0.99

A

+-2.58

75
Q

Interpreting confidence intervals in terms of probabilistic behavior of the interval

A

“There is a 95% probability that the interval [a, b] will contain the population parameter”

76
Q

Hypothesis

A

Statement that some population parameter is equal to a particular value or lies in some set of values

77
Q

Type 1 error

A

Rejecting the null hypothesis when in fact it is true

78
Q

Type 2 error

A

Failing to reject the null when it is in fact false

79
Q

5 steps of hypothesis test

A
  1. State the hypotheses - the Null and Alternative
  2. Construct a ‘test’ statistic:
    Z = ((sample statistic) - (hypothesised population parameter))/SE of the sample statistic
  3. State the sampling distribution of the test statistic under the provisional assumption that the Null is true Z~N(0,1)
  4. Use the SN distribution to control the probability of a Type 1 error
  5. Make a Decision - reject or fail to reject the Null
80
Q

Time series data

A

Sequence of data points recorded in chronological order. Observations are often taken at equally-spaced points in time
Aims are (1) provide a simple model of the evolution of a variable as an aid to understanding (2) to provide a basis for forecasting/prediction
Data are not independent

81
Q

Example of how to linearize time series data

A

Exponential growth —> take their logarithm

82
Q

Breaks

A

Variations that occur due to sudden causes and are usually ex ante unpredictable

83
Q

Seasonality

A

Predictable periodic pattern that reoccurs or repeats over regular intervals

84
Q

Cycles

A

A series follows an up and down pattern that is not seasonal

85
Q

Deterministic time series

A

One which can be expressed explicitly by an analytic expression, it has no probabilistic or random aspects

86
Q

Stochastic time series

A

Non-deterministic time series is one which cannot be described by an analytic expression
Reasons for randomness (1) all the information necessary to describe it explicitly is not available, although it might be in principle or (2) the nature of the generating process is inherently random

87
Q

Stationarity

A

The idea that there is nothing statistically special about the segment of history that you observed in the sense that the statistical properties of the process generating the data are invariant to shifts in the window of observation

88
Q

Strong stationarity

A

All statistical features of a distribution are invariant to time-shifts

89
Q

Weak stationarity

A

E[Xt] and Var(Xt) do not vary with time
Cov(Xt, Xt-h) and Corr(Xt, Xt-h) do not vary with time only h

90
Q

Transforming a time series to stationarity

A

1) differencing the data
2) trend -> fitting a curve to the data
3) non-constant variance -> taking logarithm or square root of the variance

91
Q

Causal effect

A

Difference between potential outcomes

92
Q

Observed difference between groups=

A

ATT+Selection Bias

93
Q

Selection bias

A

Average difference in the no treatment outcome between the treated and un-treated groups
Reflects the idea that this bias will arise if individuals are selected for treatment on the basis of potential outcomes

94
Q

Randomisation makes treatment independent of potential outcomes

A

The mean potential outcomes are identical for the treated and untreated groups
Selection bias is zero

95
Q

Internal validity

A

Findings for the sample are credible

96
Q

External validity

A

Its findings can be credibly extrapolated to the population or Real World policy of interest

97
Q

Threats to internal validity

A

1) Contamination: People in the control group access the treatment anyway
2) Non-compliance: individuals who are offered the treatment refuse to take it
3) Hawthorne Effect: a phenomenon in which participants alter their behaviour as a result of being part of an experiment or study
4) Placebo effect: the placebo effect impacts outcomes because of perceived changes

98
Q

Threats to external validity

A

1) small/local nature of RCT (geographic area, institutional environment, demographic group)
2) spillover effects
3) short durations —> don’t know long term impact

99
Q

Conditional Independence Assumption

A

Assignment to treatment id independent of potential outcomes conditional on covariates

100
Q

Problems with conditional independence

A

1) credibility of CIA (more factors)
2) the common support problem/curse of dimensionality - few or no observations for certain groups
3) “bad controls” - controlling for variables which are themselves outcomes

Prelims (6 decks)

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