Continuous random variable; pdf properties

Continuous random variable: a random variable that can take any value in an interval (or in the whole real line), so its possible values are uncountable. Its probabilities are found as areas under a curve, not from individual values; P(X = a) = 0 for every single value a.

Example: the time taken (in minutes) by a student to load a web page, the life of a bulb in hours, or the height of a student. If X = life of a bulb, X can be 1000.5 h, 1000.57 h, and so on.

Probability density function (pdf)

A function f(x) used to describe a continuous random variable, so that P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx.

Properties of pdf

1. f(x) ≥ 0 for all x (a density is never negative).
2. Total area under the curve is one: ∫₋∞^∞ f(x) dx = 1.
3. P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx = area between a and b.
4. P(X = a) = ∫ₐᵃ f(x) dx = 0, so P(a < X < b) = P(a ≤ X ≤ b) = P(a ≤ X < b) = P(a < X ≤ b).
5. f(x) itself is not a probability and may exceed 1; only the area is a probability.

Distribution function (cdf)

F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt

Properties of distribution function

1. 0 ≤ F(x) ≤ 1 for all x.
2. F(x) is non-decreasing: if a < b then F(a) ≤ F(b).
3. F(−∞) = 0 and F(+∞) = 1.
4. F(x) is continuous (for continuous X) and P(a < X ≤ b) = F(b) − F(a).
5. f(x) = dF(x)/dx wherever the derivative exists.

Example: for f(x) = 2x, 0 ≤ x ≤ 1: F(x) = x², and P(0.5 < X < 1) = F(1) − F(0.5) = 1 − 0.25 = 0.75.