### 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 (C.4)

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 (PDF): (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
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