Geometric probability, density and continuous random variables

This studio follows the Russian course: geometric probability (a random point in a figure, segment or arc) in grade 9, continuous random variables and density in grade 11, and the uniform and exponential distributions at advanced level.

For a point chosen at random, the probability of a part is the measure of the part divided by the measure of the whole: length, arc length or area. For a continuous variable a single exact value has probability 0, so we use intervals; the probability is the area under the density graph, and the total area is 1.

Uniform on [a, b]: f(x) = 1 ÷ (b − a). Exponential (waiting time): f(x) = λe^(−λx) for x ≥ 0, with mean 1 ÷ λ. Common mistakes: comparing sides instead of areas, reading the density as a probability, mixing up λ and the mean time.