Summary: Random variables.
Definition. We say
F : R [0, 1]
is a cumulative distribution function if
(i) x < y F(x) F(y);
(ii)limx- F(x) = 0
(iii) limxa F(x) = F(a) whenever a R; and
(iv) limxF(x) = 1.
We say
p : R [0, 1]
is a probability mass function if
(i) {x R : p(x) = 0} is countable and
(ii) xR p(x) = 1.
We say
f : R [0, )
is a probability density function if
(i) f is integrable and
(iii)