### Random Variable

**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) |

This may be called a discrete

*triangular*distribution. It can be shown to be given by the

*convolution*of the discrete uniform distribution for one die with itself. This is a general fact for sums of random variables (the distribution of the sum equals the convolution of the component distributions).

**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) |

Note, however, that the probability of any

*single*angle is zero. This is our first example of a

*continuous probability distribution*. Therefore, we cannot simply define the probability of outcome for each . Instead, we must define the

*probability density function*(

(C.8) |

To calculate a probability, the PDF must be

*integrated*over one or more

*intervals*. As follows from Lebesgue integration theory (``measure theory''), the probability of any countably infinite set of discrete points is zero when the PDF is finite. This is because such a set of points is a ``set of measure zero'' under integration. Note that we write for discrete probability distributions and for PDFs. A discrete probability distribution such as that in (C.4) can be written as

(C.9) |

where denotes an

*impulse*.

^{C.1}

**Next Section:**

Stochastic Process

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Independent Events