Cumulative distribution function

The cumulative distribution function is obtained by accumulating probability from the left. For a discrete variable, \(F(x)=\sum_{t\le x}p(t)\) is a staircase that jumps at each possible value; for a continuous one, \(F(x)=\int_{-\infty}^{x}f(t)\,dt\) is a curve without jumps. With \(f(x)=2x\) on \([0,1]\) we get \(F(x)=x^2\) on that interval.

Every \(F\) has three properties: it never decreases, tends to 0 as \(x\to-\infty\) and to 1 as \(x\to+\infty\); it is also right-continuous. Conversely, any function with these properties is the distribution of some variable. This gives a quick way to show that a candidate is not valid.

With it, interval probabilities become subtractions: \(P(a<X\le b)=F(b)-F(a)\) and \(P(X>a)=1-F(a)\). The jumps of \(F\) are the masses, \(P(X=a)=F(a)-F(a^-)\), and where \(F\) is differentiable the density is \(f(x)=F'(x)\). A common mistake is confusing \(P(X>a)\) with \(F(a)\), or adding instead of subtracting.

Quantiles are read by inverting \(F\): the value \(m\) with \(F(m)=\tfrac12\) is the median. The expectation that comes next in the course can be computed from either the mass or the density, and \(F\) is the thread connecting both descriptions.