D2. Data and Variables

Categorical data is

Qualitative: often binary indicators

Ordinal: order is meaningful but differences aren’t

Numerical (Quantitative) data is

Cardinally meaningful, i.e. discrete or continuous

The support of a variable is

The set of values it can take

Multivariate variables are

The list of measured features when we measure several features of an object

Cross-sectional data sets have

One observation per unit
E.g. data on one attribute measured in N people a cross-section dataset would be indicated as

A time series is

A series of data points indexed in time order

A time series on a multivariate variable is

A single observation

The function for discrete variables is a

Probability Mass Function

The function for continuous variables is a

Probability Density Function

A PDF for a continuous distribution with a = 1.5 and b = 2 looks like

A PMF looks like

The formal definition of a PDF is

What are the parameters of the binomial distribution

N and p

What does the PMF of a binomial distribution with N = 10 and p = 0.5 look like relative to N = 15 and N = 20 (same p)

All PDFs have the properties

They are non-negative: f (x) ≥ 0

The area underneath them is equal to one: ∫ f(x)dx = 1

The PDF for X~U (a, b) is

f(x) = 1/(b-a) if a≤x≤b

=0 otherwise

The cumulative distribution function (CDF) is

A function giving the probability that the variable X is less than or equal to some value x

The CDF is defined mathematically as

F(x) = P(X≤x) for all x in the support of X

For a variable X~U(a, b) the CDF is

For any CDF

In the CDF, the 1st quartile is the value such that

P(X ≤ x) = 0.25

The probability that the value of a Standard Normal variable would lie within ±1.645 of its central location (μ = 0) is

0.9

The probability that the value of a Standard Normal variable would lie within ±1.96 of its central location (μ = 0) is

0.95

The probability that the value of a Standard Normal variable would lie within ±2.58 of its central location (μ = 0) is

0.99

if X ∼ N(μ, σ^2) and Z = (X−μ)/σ then

Z ∼ N(0, 1)

Suppose X ∼ N(μ, σ^2) and further suppose that we want to know P(X ≤ x). What value do we look up in the statistical table?

The PDF of a variable x, given the CDF, is

The joint CDF is

F (x, y) = P (X ≤ x, Y ≤ y)

The joint PMF is

f (x, y) = P (X = x, Y = y)

The joint PDF is

The marginal PDF of x, f(x), is

f(x, whatever the value of y)

The PDF of a variable x, given the CDF, is

If X and Y are independent then the joint PDF is

The product of the marginals. f (x, y) = f (x) g (y)

If X and Y are independent then the joint CDF is

The product of the marginals. F (x, y) = F (x) G (y)