 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability/probability/9/a/i/solution

The cumulative distribution function is $F_{X} (x) = P (X \leq x)$ . In the absolutely continuous setting of a probability density function, the probability density function is a nonnegative integrable function

f_{X}

satisfying

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

Thus

P (a < X \leq b) = \int_{a}^{b} f_{X} (t) d t

and

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

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