We say that the sample space of a random experiment is infinite if there are an infinite number of possible outcomes of that random experiment. The number of infinite outcomes can be of two types – countably infinite or uncountably infinite depending on whether the random variable is discrete or continuous.
Examples of countably infinite sample spaces:
By countably infinite sample space we mean that it is possible to put the possible outcomes in a sequence and count them in a step-by-step manner. Some discrete random variables such as the Poisson distribution and the geometric distribution have a countably infinite number of possible outcomes. Some examples of countably infinite sample spaces are:
- Number of typographical errors in a book. We can have 0 errors, 1 errors, 2 errors, so on in the book. Theoretically speaking there can be any large number of errors in the book up to infinity.
- Suppose an archer wants to hit a mark. Let X be the number of tries the archer misses before hitting the mark. Then X can be equal to 0 if the archer hits the mark at the first try. The value of X can be 1 if the archer hits successfully at the second try. The value of X can be 100 if the archer misses for the first 100 times and successfully hits at the 101st trial. This means than X can take any value among the whole numbers up to infinity.
Examples of uncountably infinite sample spaces:
By an uncountably infinite sample space, we mean that the number of possible outcomes is so large that it is impossible to count them in a sequential manner. For instance, the set of real numbers is an uncountably infinite set as we cannot predict the number that comes after any given number. All continuous random variables such as the uniform distribution, exponential distribution, and normal distribution have an uncountably infinite number of possible outcomes. Some examples of uncountably infinite sample spaces are:
- Let X denote the weight of a person randomly chosen in a particular city. The X takes uncountably infinite values since the weight of the person could be 75kgs or 75.75 kgs or 75.12 kgs or 75.10 kgs, etc. The possible values of X are too many to explicitly ennumerate and there is a spectrum of possible values for X. The random variable X is this case can be modelled using the normal distribution.
- Suppose a person is waiting for a train at the railway station. Let X denote the waiting time of the person. Then X takes uncountably many values since the person may have to wait for 1 minute or 1.5 minutes or 1.75 minutes or 1.995 minutes, etc. Clearly there are too many possible cases and counting them is not possible. The waiting time can be modelled using the exponential distribution.
