### 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