Engineering Analysis/Random Variables

Random Variables
A random variable is a variable that takes a random value at any particular point t in time. The properties of the random variable are known as the distribution of the random variable. We will denote random variables by the abbreviation "r.v.", or simply "rv". This is a common convention used in the literature concerning this subject.

Probability Function
The probability function, P[], will denote the probability of a particular occurrence happening. Here are some examples:


 * $$P[X < x]$$, the probability that the random variable X has a value less than some variable x.
 * $$P[X = x]$$, the probability that the random variable X has a value equal to some variable x.
 * $$P[X < x, Y > y]$$, the probability that the random variable X has a value less than x, and the random variable Y has a value greater than y.

Example: Fair Coin
Consider the example that a fair coin is flipped. We will define X to be the random variable, and we will define "head" to be 1, and "tail" to be 0. What is the probability that the coin is a head?


 * $$P[X = 1] = 0.5$$

Example: Fair Dice
Consider now a fair 6-sided dice. X is the r.v., and the numerical value on the face of the die is the value that X can take. What is the probability that when the dice is rolled, the value is less than 4?


 * $$P[X < 4] = 0.5$$

What is the probability that the value will be even?


 * $$P[X\mbox{ is even}] = 0.5$$

Notation
We will typically write random variables as upper-case letters, such as Z, X, Y, etc. Lower-case letters will be used to denote variables that are related with the random variables. For instance, we will use "x" as a variable that is related to "X", the random variable.

When we are using random variables in conjunction with matrices, we will use the following conventions:


 * 1) Random variables, and random vectors or matrices will be denoted with letters from the end of the alphabet, such as W, X, Y, and Z. Also, &Theta; and &Omega; will be used as a random variables, especially when we talk about random frequencies.
 * 2) A random matrix or vector, will be denoted with a capital letter. The entries in that random vector or matrix will be denoted with capital letters and subscripts. These matrices will also use letters from the end of the alphabet, or the Greek letters &Theta; and &Omega;.
 * 3) A regular coefficient vector or matrix that is not random will use a capital matrix from the beginning of the alphabet, such as A, B, C, or D.
 * 4) Special vectors or matrices that are derived from random variables, such as correlation matrices, or covariance matrices, will use capital letters from the middle of the alphabet, such as K, M, N, P, or Q.

Any other variables or notations will be explained in the context of the page where it appears.

Conditional Probability
A conditional probability is the probability measure of one event happening given that another event already has happened. For instance, what are the odds that your computer system will suddenly break while you are reading this page?


 * $$P[\mbox{computer breaks}] = \mbox{small}$$

The odds that your computer will suddenly stop working is very small. However, what are the odds that your computer will break given that it just got struck by lightning?


 * $$P[\mbox{computer breaks}|\mbox{struck by lightning}] = \mbox{large}$$

The vertical bar separates the things that haven't happened yet (the a priori probabilities, on the left) from the things that have already happened and might affect our outcome (the a posteriori probabilities, on the right). As another example, what are the odds that a dice rolled will be a 2, assuming that we know the number is less than 4?


 * $$P[X = 2|X < 4] = 0.33$$

If X is less than 4, we know it can only be one of the values 1, 2, or 3. Or another example, what if a person asks you "I'm thinking of a number between 1 and 10", what are your odds of guessing the right number?


 * $$P[X = x|0 < X < 10] = 0.1$$

Where x is the correct number that you are trying to guess.