 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability-resit/9/a/i/solution

F_{X} (x) = P (X \leq x) .

f_{X}

with

F_{X} (x) = \int_{- \infty}^{x} f_{X} (u) d u, \int_{- \infty}^{\infty} f_{X} (u) d u = 1 .

Thus interval probabilities are integrals of

f_{X}

, and

F_{X}^{'} = f_{X}

f_{X}

The word “continuous” in the source is understood in this density sense. A merely continuous cumulative distribution function need not possess a probability density function; it can instead describe a singular continuous distribution.

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