Definition:
A random variable
is defined as a real- or complex-valued
function of some random event, and is fully characterized by its
probability distribution.
Example:
A random variable can be defined based on a coin toss by defining
numerical values for heads and tails. For example, we may assign 0 to
tails and 1 to heads. The probability distribution for this random
variable is then
Example:
A die can be used to generate integer-valued random variables
between 1 and 6. Rolling the die provides an underlying random event.
The probability distribution of a fair die is the
discrete uniform distribution between 1 and 6. I.e.,
(C.5) |
Example:
A pair of dice can be used to generate integer-valued random
variables between 2 and 12. Rolling the dice provides an underlying
random event. The probability distribution of two fair dice is given by
(C.6) |
Example:
Consider a random experiment in which a sewing needle is dropped onto
the ground from a high altitude. For each such event, the angle of
the needle with respect to north is measured. A reasonable model for
the distribution of angles (neglecting the earth's magnetic field) is
the continuous uniform distribution on
, i.e., for
any real numbers
and
in the interval
, with
, the probability of the needle angle falling within that interval
is
(C.7) |
(C.8) |
(C.9) |