Statistics and Probability: Random Variables

Flashcards on Random Variables and Probability Distributions

Random Variable

A function that associates a real number with each element in the sample space.

X and x

Denotes a random variable (uppercase), represents one of its values (lowercase).

Remark on Random Variables

Each outcome in the sample space must be mapped to exactly one real number.

Discrete Sample Space

A sample space containing a finite number of possibilities or an unending sequence with as many elements as there are whole numbers.

Discrete Random Variable

A random variable defined over a discrete sample space.

Bernoulli Random Variable

A random variable where 0 and 1 describe the possible values.

Continuous Sample Space

A sample space containing an infinite number of possibilities equal to the number of points on a line segment.

Continuous Random Variable

A random variable defined over a continuous sample space.

Probability Function/Mass Function/Distribution (Discrete)

Set of ordered pairs (x, f(x)) where f(x) >= 0, sum of f(x) = 1, and P(X=x) = f(x).

Cumulative Distribution Function (Discrete)

F(x) = P(X <= x) = Sum of f(t) for t <= x.

Probability Density Function (PDF) - Continuous

A function f(x) where f(x) >= 0 for all x, the integral from -inf to inf of f(x) dx = 1, and P(a < X < b) = integral from a to b of f(x) dx.

Cumulative Distribution Function (Continuous)

F(x) = P(X <= x) = integral from -inf to x of f(t) dt.