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

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- Solution i

Solution

The cumulative distribution function is

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

For an absolutely continuous probability distribution, the probability density function is a nonnegative integrable function

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}

almost everywhere, in particular at continuity points of

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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