The sample space of a random variable is the set of all possible outcomes of the random variable. An event is a subset of the sample space. A probability model assigns probabilities to elements in the sample space.
Examples
- A researcher might be interested in the behavior of the three-month Treasury bill interest rate. The random variable is the rate on the three-month Treasury bill. The sample space is the set of all possible values of the Treasury bill rate, [0, 8). An example of an event is the set of interest rates between 3% and 5%, [0.03,0.05].
- Suppose we roll a six-sided die until we observe the number 6. One random variable for this experiment is the number of times we roll the die. Define X = number of die rolls until a six is observed. The sample space is the set of integers [1,2,...,∞]. One event is X = {1}; in English, this is the event that we see a six on the first roll of the die. Another event is X = {2,4,6}, which is the event that we roll the first six on the 2nd, 4th or 6th roll of the die. The probability model is given by P(X=k) = (5/6)(k-1)*(1/6).