## 3    Random Variables and Discrete Distributions

### 3.1 Random Variables

Def: A random variable X is a variable whose value depends on chance, i.e., whose value depends on the outcome of some experiment.
• use capital letters to denote random variables
ex:
Roll 2 dice, and let X = sum of values on faces.
• X is a random variable: its value depends on the outcome of the roll of the dice.
• the values that X can take are 2, 3, 4, ..., 12.
• since this is a discrete set of values, X is called a discrete random variable
ex:
Let R = number of inches of rainfall received at Allentown airport on given day
• R can take on any value in the interval [0, 10] (for example, 3.0", 1.257", etc.)
• since there is a continuum of possible values, R is called a continuous random variable
ex:
Keep flipping a fair coin until you get a tail; let N = number of flips.
• N can take on any of the values  1, 2, 3, 4, 5, ...
• N is a discrete random variable
More formally:
• A random variable is a function whose domain is the sample space of some random experiment: the value the random variable takes on is determined by the outcome of the experiment.
• A random variable is discrete if its range (the set of values which it can take on) is countable, i.e., either finite or countably infinite, and is continuous otherwise.