Continuous Probability Densities

A continuous random variable is a theoretical representation of a continuous variable such as height, mass or time. A **continuous random variable** is specified by its probability density function (**p.d.f.**) written as f(x). If X is a continuous random variable with probability density function f(x) valid over the range a <= x <= b, then - f(x) >= 0 - _all x f(x), dx = 1, i.e, _ab f(x), dx = 1. - P(X <= x_1) = _ax_1 f(x), dx, where a <= x_1 <= x_2 <= b. - P(X >= x_2) = _x_2b f(x), dx, where a <= x_1 <= x_2 <= b. - P(x_1 <= X <= x_2) = _x_1x_2 f(x), dx, where a <= x_1 <= x_2 <= b. **Note:** P(x_1 <= X <= x_2) = P(X = x_1) + P(x_1 < X < x_2) + P(X = x_2) = 0 + P(x_1 < X < x_2) + 0 = P(x_1 < X < x_2). P(x_1 <= X <= x_2) = P(x_1 < X < x_2) = P(x_1 <= X < x_2) = P(x_1 < X <= x_2).