Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii
- iii a
  - Solution iii

i

Solution

The cumulative distribution function is

F_{X} (x) = P (X \leq x) = \int_{- \infty}^{x} f (t) d t .

The total integral of the nonnegative probability density function is one.

ii

Solution

For a random variable with a probability density function, a singleton has zero probability:

P (X = x) = \int_{{x}} f (t) d t = 0.

Equivalently, its cumulative distribution function is continuous, so it has no jump at

x

.

iii

Solution

The integers form a countable set, and the events

{X = k}

are disjoint. By countable additivity and part ii,

P (X \in Z) = k \in Z \sum P (X = k) = 0.

Countability is essential: one cannot use the same sum argument over an arbitrary uncountable collection of zero-probability events.

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