Random Variable: Process generating random outcomes.

Sample: Outcomes from a random variable.

Descriptive Statistics: Summary of a sample.

Inferential Statistics: Educated guess about the random variable that generated a sample.

Process that generates random outcomes.

Tied to a population.

Has possible outcomes defined by a sample space.

Is fundamentally defined by a probability distribution.

A population is the entire set of things under study.

E.g., all people in the world, all possible people in the world, all the neurons in the universe, the set of all possible action potential times relative to stimulus onset etc.

The set of all possible outcomes of a random variable.

E.g., the set of all possible rolls of a die is \(\{1, 2, 3, 4, 5, 6\}\). It is discrete and finite.

E.g., the set of all possible action potential times relative to stimulus onset spans the interval \((0, \infty)\). It is continuous and infinite.

Describes the likelihood of each outcome in the sample space.

Discrete: A function that gives the probability of each outcome.

Continuous: A function that gives the probability density for each outcome.

The sample space goes on the x-axis.

The probability of each outcome goes on the y-axis.

Consider a biased coin with PMF \(P(X = x) = 0.3\) for heads and \(P(X = x) = 0.7\) for tails.

Samples obtained from this distrubtion are random sequences of heads and tails.

Because the coin is biased, we expect to see more tails than heads.

Random variables and probability distributions are clean and idealized.

Samples from random variables are… well… random.

We use samples from random variables to estimate their probability distributions.

For discrete random variables, we use relative frequency bar plots.

For continuous random variables, we use relative frequency histograms or density plots.