Probability Distribution of a Continuous Random Variable

A continuous random variable behaves quite differently from a discrete one: its possible values fill an entire interval of real numbers, so there are uncountably infinitely many of them, and no single value can be given a positive probability. A typical example is the time an athlete takes to complete a 1000-metre race — a genuinely continuous measurement. Everything learned about a discrete random variable's probability distribution is now extended to the continuous case with the help of two tools: the probability density function (p.d.f., section 7.4.1) and the cumulative distribution function (c.d.f., section 7.4.2). If the possible values of a continuous random variable $X$ fill an interval $[a, b]$, that interval is called the support of $X$, usually also denoted $S$. …