## 3.2 Discrete random variables

Recall from Topic 1 that a discrete variable is one where "the set of all possible values is countable. For example, the number of heartbeats per minute, or the number of heads observed when flipping a coin five times". Consequentially, number of heartbeats per minute, and number of heads observed when flipping a coin five times, are both examples of variables that are discrete random variables.

Consider the following: let $$X$$ denote the number of heads observed when flipping a coin five times. Then the possible values for $$X$$ are $$0, 1, 2, 3, 4, 5$$.

Or: let $$X$$ denote the number of heartbeats per minute of university students. Then the possible values for $$X$$ are $$0, 1, 2, 3, \ldots$$.

In the first example above, we were able to list every single possible outcome for $$X$$. In the second example, were not able to list every possible outcome for $$X$$ because it would not make sense to place an upper limit on the number of heartbeats per minute. However, what both examples had in common was that the possible values (or outcomes) for $$X$$ could be written down sequentially with a recognisable pattern. In other words, they are countable. This means $$X$$ as defined in both of these examples was a discrete random variable.

The concepts of mean, variance, and standard deviation can be related to a discrete random variable $$X$$ in exactly the same way as we saw for a continuous random variable $$X$$. That is:

Expected value, variance and standard deviation of a random variable

• The expected value (or mean) of $$X$$ can be denoted $$\text{E}(X) = \mu$$
• The variance of $$X$$ can be denoted $$\text{Var}(X) = \sigma^2$$
• The standard deviation of $$X$$ is the square root of the variance and can be denoted $$\text{SD}(X) = \sqrt{\sigma^2} = \sigma$$