Sample Space
Consider an experiment (e.g. tossing a coin, or rolling a die).
Sample space - set of all possible outcomes.
Example:
Experiment consisting of rolling a die once.
Sample space = {1, 2, 3, 4, 5, 6}
Event
Consider an experiment (e.g. tossing a coin, or rolling a die).
Event - a subset of the sample space.
Example:
Experiment consisting of rolling a die once, the possible events:
- {2, 3},
- {6},
- empty set {} (often denoted by Φ)
- the entire sample space {1, 2, 3, 4, 5, 6}
Example
(Sample Space & Event)
Question: Consider the experiment to be tossing a coin. What is the Sample Space? What are the events associated with this Sample Space?
Answer: Notice that although the sample space includes only the outcome of the experiments which are Head (H) and Tail (T), the events associated with this samples space includes all subsets of the state space which include also the empty set which in this case is the event { H n T } and the entire sample space which in this case is the event {H u T}.
Mutually Exclusive & Exhaustive
Events are called mutually exclusive if their intersection is the empty set.
A set of events is exhaustive if its union is equal to the sample space.
Examples
(Mutually Exclusive & Exhaustive)
Example 1: tossing a coin only once
- The events {H} (Head) and {T} (Tail) are both mutually exclusive and exhaustive.
- What is the sample space (the set) of all possible events in this case?
Example 2: rolling a die only once
- The events {1}, {2}, {3}, {4}, {5}, and {6} are both mutually exclusive and exhaustive.
- The events {4}, {5}, and {6} are mutually exclusive but are not exhaustive.
Random Variable
A random variable is a real valued function defined on the sample space.
This function X = X(ω) assigns a number to each outcome ω of the experiment.
Example:
tossing a coin experiment
X = 1 for Head {H}
X = 0 for Tail {T}
Note that the function X is deterministic (not random), but the ω is unknown before the experiment is performed. Therefore X(ω) is called a random variable.
If X is a random variable then Y=g(X) for some function g is also a random variable.
Examples:
- Y = cX for some scalar c is a random variable.
- Y = Xn for some integer n is a random variable.
- If X1, X2, X3 , ... , Xn is a sequence of random variables, then 
is also a random variable.
Probability
Consider a sample space S. Let A be a subset of S. The probability of A is the function on S and all its subsets, denoted P(A) that satisfies the following three axioms:
1. 0 ≤ P(A) ≤ 1
2. P(S) = 1
3. The probability of the union of mutually exclusive events is equal to the sum of the probabilities of these events.
Text / Reference Books
- Moshe Zukerman, Introduction to Queueing Theory and Stochastic Teletraffic Models (Chapter 1)
http://www.ee.cityu.edu.hk/~zukerman/classnotes.pdf
- D. Bertsekas and J. N. Tsitsiklis, Introduction to Probability, Athena Scientific, Belmont, Massachusetts 2002.
- S. M. Ross, A first course in probability, Macmillan, New York, 1976.
