1.1 Random Variables
In many areas of science we are interested in quantifying the probability that a certain outcome of an experiment occurs. We can use a random variable to identify numerical events that are of interest in an experiment. In this way, a random variable is a theoretical representation of the physical or experimental process we wish to study. More precisely, a random variable is a quantity without a fixed value, but which can assume different values depending on how likely these values are to be observed; these likelihoods are probabilities.
To quantify the probability that a particular value, or set of values (called an event), occurs, we use a number between 0 and 1. A probability of 0 implies that the event cannot occur, whereas a probability of 1 implies that the event must occur. Any value in the interval (0, 1) means that the event will only occur some of the time. Equivalently, if an event occurs with probability p, then this means there is a p(100)% chance of observing this event.
Conventionally, we denote random variables by capital letters, and particular values that they can assume by lowercase letters. So we can say that X is a random variable that can assume certain particular values x with certain probabilities.
We use the notation Pr(X = x) to denote the probability that the random variable X assumes the particular value x. The range of values x for which this expression makes sense is of course dependent on the possible values of the random variable X. We distinguish between two key cases.
If X can assume only finitely many or countably many values, then we say that X is a discrete random variable. Saying that X can assume only finitely many or countably many values means that we should be able to list the possible values for the random variable X. If this list is finite, we can say that X may take any value from the list x1, x2,..., xn, for some positive integer n. If the list is (countably) infinite, we can list the possible values for X as x1, x2,.... This is then a list without end (for example, the list of all positive integers).
We summarize the basic notions of a discrete random variable:
- A discrete random variable X is a quantity that can assume any value x from a discrete list of values with a certain probability.
- The probability that the discrete random variable X assumes the particular value x is denoted by Pr(X = x). This collection of probabilities, along with all possible values x, is the probability distribution of the random variable X.
- A discrete list of values is any collection of values that is finite or countably infinite (i.e. can be written in a list).
This terminology is in contrast to a continuous random variable, where the values the random variable can assume are given by a continuum of values. For example, we could define a random variable that can take any value in the interval [1,2]. The values X can assume are then any real number in [1,2]. We will discuss continuous random variables in detail in the second chapter. For now, we deal strictly with discrete random variables.
We state a few facts that should be intuitively obvious for probabilities in general. Namely, the chance of some particular event occurring should always be nonnegative and no greater than 100%. Also, the chance that something happens should be certain. From these facts, we can conclude that the chance of witnessing a particular event should be 100% less the chance of seeing anything but that particular event.
|Discrete Probability Rules|
1. Probabilities are numbers between 0 and 1 inclusive: 0 ≤ Pr(X = xk) ≤ 1 for all k
2. The sum of all probabilities for a given experiment (random variable) is equal to one:
3. The probability of an event is 1 minus the probability that any other event occurs:
Example: Tossing a Fair Coin Once
If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the tossed coin is a "fair" coin, it is equally likely that the coin will land as tails or as heads. In other words, there is a 50% chance (1/2 probability) that the coin will land heads, and a 50% chance (1/2 probability) that the coin will land tails. Notice that the sum of these probabilities is 1 and that each probability is a number in the interval [0,1].
We can define the random variable X to represent this coin tossing experiment. That is, we define X to be the discrete random variable that takes the value 0 with probability 1/2 and takes the value 1 with probability 1/2. Notice that with this notation, the experimental event that "we toss a fair coin and observe heads" is the same as the theoretical event that "the random variable X is observed to take the value 0"; i.e. we identify the number 0 with the outcome of "heads", and identify the number 1 with the outcome of "tails". We say that X is a Bernoulli random variable with parameter 1/2 and can write X ~ Ber(1/2).
Example: Tossing a Fair Coin Twice
Similarly, if we toss a fair coin two times, there are four possible outcomes. Each outcome is a sequence of heads (H) or tails (T):
Because the coin is fair, each outcome is equally likely to occur. There are 4 possible outcomes, so we assign each outcome a probability of 1/4.
Equivalently, we notice that for any of the four possible events to occur, we must observe two distinct events from two separate flips of a fair coin. So for example, to observe the sequence HH, we must flip a fair coin once and observe H, then flip a fair coin again and observe H once again. (We say that these two events are independent since the outcome of one event has no effect on the outcome of the other.) Since the probability of observing H after a flip of a fair coin is 1/2, we see that the probability of observing the sequence HH should be (1/2)×(1/2) = 1/4.
Observe that again, all of our probabilities sum to 1, and each probability is a number on the interval [0, 1]. Just as before, we can identify each outcome of our experiment with a numerical value. Let us make the following assignments:
- HH -> 0
- HT -> 1
- TH -> 2
- TT -> 3
This assignment defines a numerical discrete random variable Y that represents our coin tossing experiment. We see that Y takes the value 0 with probability 1/4, 1 with probability 1/4, 2 with probability 1/4, and 3 with probability 1/4. Using our general notation to describe this probability distribution, we can summarize by writing
Pr(Y = k) = 1 / 4, for k = 0,1,2,3.
Notice that with this notation, the experimental event that "we toss two fair coins and observe first tails, then heads" is the same as the theoretical event that "the random variable Y is observed to take the value 2". We say that Y is a uniform discrete random variable with parameter 4 since Y takes each of its four possible values with equal, or uniform, probability. To denote this distributional relationship, we can write Y ~ Uniform(4).