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Continuous Learning > Statistics/Probability > Flashcards

Statistics/Probability Flashcards

1
Q

sample space

A

the set of all possible sample points for an experiment, e.g. S={HH,TT,HT,TH} for two times head tails flip

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2
Q

dependent events regarding probability

A

e.g. picking marbles out a bag

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3
Q

Covariance

A
  • When calculated between two variables, X and Y, it indicates how much the two variables change together.
  • Cov(X,Y)=E[(X−EX)(Y−EY)] = E[XY]−(EX)(EY)
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4
Q

P–P plot

A

probability–probability plot or percent–percent plot or P value plot: probability plot for assessing how closely two data sets agree, or for assessing how closely a dataset fits a particular model.

It works by plotting the two cumulative distribution functions against each other; if they are similar, the data will appear to be nearly a straight line.

For input z the output is the pair of numbers giving what percentage of f and what percentage of g fall at or below z.

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5
Q

Q–Q plot

A

quantile–quantile plot: for comparing two probability distributions by plotting their quantiles against each other. A point (x, y) on the plot corresponds to one of the quantiles of the second distribution (y-coordinate) plotted against the same quantile of the first distribution (x-coordinate).

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6
Q

PMF (Probability Mass Function)

A

A probability mass function (PMF) is a mathematical function that calculates the probability that a discrete random variable will be a specific value. It assigns a particular probability to every possible value of the variable.

Table: With each row an outcome + probability

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7
Q

the conditional probability for a cancel given snow.

A

P(Cancel∣Snow), the ∣ is short for ‘given’

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8
Q

An event happens independently of a condition if

A

P(event∣condition)=P(event)

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9
Q

Kolmogorov-Smirnov (K-S) Test

A

non-parametric test that compares the empirical distribution of the data with a theoretical distribution.

It helps determine how well the theoretical distribution fits the data.

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10
Q

K-S Statistic

A

The K-S statistic measures the maximum distance between the empirical cumulative distribution function (ECDF) of your data and the cumulative distribution function (CDF) of the theoretical distribution.

In simpler terms, it quantifies the biggest difference between what you observed (your data) and what you would expect if the data followed the theoretical distribution.

The K-S statistic ranges from 0 to 1:
A smaller K-S statistic indicates that the empirical distribution is very close to the theoretical distribution.

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11
Q

outcome = model + error –> how are the parts called?

A

model = systematic part, error = unsystematic part

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12
Q

Descriptive Statistics

A

collect, organize, display, analyze, etc.

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13
Q

Inference Statistics

A
  • Predict and forecast values of population
    parameters
  • Test hypothesis and draw conclusions about values
    of population parameters
  • Make decisions
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14
Q

Central Tedency

A

1st moment - mean, median, mode

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15
Q

Spread

A

2nd moment - MAD, Variance, SD, coefficient of variation (CV = SD/mean), range, IQR

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16
Q

Skweness

A

3rd moment - measure of asymmetry, positive skew (tail pointing to high values (body of the distribution is to the left), negative skew

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17
Q

Kurtosis

A

4th moment - Measure of heaviness of the tails, leptokurtic (heavy tails), platykurtic (light tails)

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18
Q

Which kurtosis has a normal distribution?

A

3 (mesokurtic)

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19
Q

statistical test on prices vs returns:

A

prices are not predictable, returns are predictable (they are “stationary”)

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20
Q

Standard Error calculation

A

SE = SD / (n^1/2)

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21
Q

Sample standard deviation

A

𝑠

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21
Q

Population standard deviation

A

𝜎 (sigma)

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22
Q

Central Limit Theorem

A

states that: the distribution of sample mean, 𝑋ത, will approach a Normal distribution as sample size 𝑛 increases (𝑛 ≥ 30)

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23
Q

Sample variance - do you use n or n-1?

A

n-1

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24
Q

Random variable:

A

𝑋

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25
Q

Cumulative Density Function of Standard Normal:

A

Φ (z)

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26
Q

Pivotal distribution

A

N(0,1)

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27
Q

Population mean - greek letter:

A

μ (mu)

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28
Q

sample standard deviation

A

s

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29
Q

Confidence interval

A

sample mean +/- z-value * (sigma or SE / root(n))

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30
Q

In the sample, you approximate mu and sigma with…

A

x (sample mean) and sample standard deviation

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31
Q

Population standard deviation

A

𝜎

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32
Q

Particular observation of a Standard Normal (also
known as ‘z-critical value’)

A

z

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33
Q

Parameter of 𝒕-distribution (also known as ‘degrees of
freedom’):

A

𝜐

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34
Q

t-critical value

A

t

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35
Q

Important: are you given sigma or s?

A

If n is < 30, but you are given sigma, you can use sigma

36
Q

t-distribution

A
  • Has thicker tails than Normal (i.e. larger chance of
    extreme events).
  • Its shape depends on a single parameter “nu” 𝜈 =
    𝑛 – 1, where n is the number of observations.
  • Assumption: t-distribution assumes that the data
    originates from a Normal Distribution.
37
Q

3 main types of distribution

A

Gaussian, Poisson, Chi-square

38
Q

Statistical stationarity:

A

A stationary time series is one whose statistical properties such as mean, variance, autocorrelation, etc. are all constant over time.

39
Q

get probability of z-value - function

A

xpnorm(probability, mean =…, sd = …)

40
Q

given a particular probability of 𝑍 < 𝑧, what is the corresponding value 𝑧?

A

qnorm(value z, mean = …, sd = …)

41
Q

Measures the amount of variability within a single
dataset - calculate SD for population and sample - comparison population vs sample SD calculation

A
42
Q

What is variance

A

the expected value of the squared deviation from the mean of a random variable

43
Q

Null Hypothesis vs Sample mean

A
  • Null Hypothesis: 𝐻0 belief about true population parameter value –> The null hypothesis will be rejected if the difference between sample means is bigger than would be expected by chance
  • Sample mean: 𝐻1 alternative
43
Q

Significance Level - letter

A

alpha –> probability of rejecting the null hypothesis when it is true

44
Q

Critical value / Cutoff point

A

𝑧 − 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 or 𝑡 − 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 –> ±z/t-value which act as cutoff points beyond which the null hypothesis should be rejected

45
Q

p-value: 𝑝

A

probability of obtaining a value of the test statistic as extreme as, or more extreme than, the actual value obtained, when the null hypothesis is true

46
Q

How to report results for statistical hypothesis testing:

A
  • “we accept the null hypothesis as truth”
  • “we cannot reject the null hypothesis”
46
Q

Hypothesis:

A

is a statement of assertion about the true value of an unknown population parameter (e.g. 𝜇 = 100)

46
Q

Null Hypothesis Test - performed using a test statistic, which is the standardised value derived from sample data, e.g. the standardized value of the sample mean

When should you use the z-statistic vs. the t-statistic in hypothesis testing?

A
46
Q

Types of Statistical Errors

A
46
Q

Conventions in Your Industry regarding alpha

A
46
Q

Calculating CI Cutoff Points
Use t-dist when 𝑛 < 30 and 𝜎 is unknown, otherwise
use 𝑧. In practice, we can use 𝑡 for all cases.

R-functions for SD Distribution and t-Distribution

A
46
Q

3 equivalent ways of testing hypothesis:

A
46
Q

Equivalent Approaches for hypothesis testing

A
47
Q

Does correlation reflect nonlinear relationships?

A

No

48
Q

True Dependent Variable and Estimated Dependent Variable

A
49
Q

True Coefficient and Estimated Coefficient

A
50
Q

Residual Error and Residual Standard Error

A
50
Q

Number of observations and nuber of independent variables

A
50
Q

Coefficient of Determination / R Squared and Adjusted R Squared

A
51
Q

Coefficient 𝒊’s Standard Error

A
51
Q

What does regression diagnostics look for?

A

Testing for “significant” relationships.

51
Q

assumed true model vs fitted model - how do you write the coefficients down for the regression equations?

A
52
Q

Different names for Y and x

A
52
Q

Fitted Model: Time-Series With Lagged Variables and Fitted Model: Autoregression - examples how they could look like:

A
53
Q

OLS minimizes…

A

the Sum of Squared Errors (SSE) with respect to regression coefficients 𝛽0, 𝛽1

54
Q

Residual Standard Error (RSE) - calculation/formula

A
  • square root of the average squared residuals
  • where 𝑛 is the number of observations and 𝑝 number of independent variables.
55
Q

What does 𝑅2 show?

A

The proportion of total variation of Y that is explained by the model (i.e. by the independent variable(s))

56
Q

𝑅2 - calculation

A
57
Q

Adjusted R2 - calculation/meaning

A
  • If 𝑛 is very large and 𝑝 is very small, the ratio is close to zero, and 𝑅2 ≈ 𝑅2adj
  • As the number of inputs (𝑥’s) increases, 𝑅2 typically increases regardless of whether the variables are useful for prediction
  • Adjusted 𝑅2 will only increase if the new 𝑥 variable improves the model more than would be expected by chance.
58
Q

linear regression modell

A

lm(y ~ x, data = name)

59
Q

Assumptions for errors in linear regression models:

A
60
Q

How is constant, time independent variance in linear regression models called?

A

homoscedasticity

61
Q

What can be conclusions of nonnormal residuals?

A

nonlinearity present, interactions between independent variables, outliers

62
Q

Possible Reasons For Systematic Errors in linear regression models:

A
  • Nonlinearity: systematic pattern in the residuals
  • Heteroscedasticity: variance of errors changes across levels of independent variable
  • Autocorrelation: errors in one period are correlated with errors in another period
63
Q

Common Nonlinear Transformations

A
  • 1/𝑥 Relationship
  • square root(𝑥) Relationship
  • x^2 Relationship
  • Exponential 𝑥^𝑏 Relationship
64
Q

We could say that the regression line has reduced our uncertainty, as measured by variances, from … to …

A
  • from the unconditional variance of s2y
  • to y s to the conditional variance of s2e
  • That is, a reduction of s2y - s2e
  • The reduction expressed as a fraction is called R-squared or R2
65
Q

A regression output usually reports the ratios between â and its standard deviation, and between bˆ and its standard deviation … which are referred to as “t-ratios”, i.e. …

A
66
Q

If the residual distribution has very “fat tails”, i.e. many more extreme values than you would like to see, it may be appropriate to think of using an alternative estimation technique, such as:

A
  • Least Absolute Value rather than Least Squares
  • This approach will weight extreme values less although a different set of diagnostics will then have to be used
67
Q

Common Nonlinear Transformations

A
  • 𝑥2 Relationship - Example: return from an investment (Y), increases quadratically (exponentially) with an increasing investment (x). This could happen due to aggressive reinvestment and compounding returns.
  • square root(𝑥) Relationship - Example: stock volatility (𝑌) increases with a decreasing rate of volume (𝑥), i.e. y = square root(x)
68
Q

What is an Interaction Term?

A
  • an independent variable in a regression model that is a product of two independent variables
  • Sometimes the partial effect of the dependent variable with respect to independent variable can depend on magnitude of yet another independent variable
69
Q

Hierarchy Principle - Interaction Term

A
70
Q

What is Multicollinearity? Effects?

A
  • appears when independent variables used inside the regression equation are highly correlated
  • Effects: fit is not improved much, additional variables add little information, may cause two or more variables to become insignificant but significance may be high if one
    variable is dropped, Parameter estimates are unreliable
71
Q

Dummy Variable:

A
  • A variable that takes on a value of 0 or 1
  • Example: 1 war year, 0 no war, treated as a reference group (usually the majority of the data in the sample)
  • When a categorical variable has k categories (we call them ‘levels), we include only k-1 dummy variables in the regression model.
  • The category that is left out is usually the one with the most frequent observations and it acts as a reference.
71
Q

Distributed lag model

A

A model for time series data in which a regression equation is used to predict current values of a dependent variable based on both the current values of an independent variable and/or its lagged (past period) values.

72
Q

Nonparametric statistics

A

statistical method in which the data are not assumed to come from prescribed models that are determined by a small number of parameters, such as the normal distribution model and the linear regression model

73
Q

RISK analysis toolbox in Excel - overview

A
74
Q

RISK build in correlation

A

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