Continuous Random Variables

Continuous random variables

A variable that can take any value in an interval (uncountably infinite)

Probability Density Function (PDF)

A function f(x) such that P(a ≤ X ≤ b) = ∫(from a to b) f(x)dx

Properties of a PDF

• f(x) ≥ 0

• ∫(from -∞ to ∞) f(x)dx = 1

Cumulative Distribution Function (CDF)

F(x) = ∫(from -∞ to x) f(t)dt

Relationship between PDF and CDF

f(x) = dF(x)/dx

Probability at a Point

For continuous variables, P(X = x) = 0

Uniform Distribution (Continuous)

f(x) = 1/(b - a) for a ≤ x ≤ b; otherwise 0

If X ~ Uniform(0, 1), then P(0.25 < X < 0.75) = 0.5

Example

Transformation of a Continuous Variable

If Y = g(X), the PDF of Y is f_Y (y) = f_X (g^-1 (y)) |d/dy g^-1 (y)|