Probability Density Function (PDF)

Study guide: Exam P-1

A Probability Density Function (PDF) describes the relative likelihood of a continuous random variable taking a particular value. The probability that X falls in an interval [a, b] is the area under f over that interval.

P(a \leq X \leq b) = \int_a^b f(x)\, dx

• The PDF must satisfy f(x) \geq 0 and \int_{-\infty}^{\infty} f(x)\, dx = 1
• P(X = x) = 0 for any single point

Probability from a Polynomial PDF If f(x) = 3x^2 for 0 \leq x \leq 1, what is P(X > 0.5)?

Answer P(X > 0.5) = \int_{0.5}^{1} 3x^2\, dx = \left[ x^3 \right]_{0.5}^{1} = 1 - 0.125 = 0.875