Discrete and Continuous Random Variables

Now that we know what a Discrete Random Variable (DRV) is, we want to compare this with what constitutes a Continuous Random Variable (CRV).

As you'll notice from the names, the only difference is the word Discrete which has been changed to Continuous. And that's the only difference in the definitions too!

Defining a Continuous Random Variable 

1. Firstly, the outcomes of the situation or the experiment must occur over an interval of space or time.
   • Remember that continuous data is numerical and is data that has been measured.
   • For example, if you were to measure the heights of students in a particular grade, the possible outcomes would exist over a range of heights, let's say between $155$155 cm and $175$175 cm. This unit of length is a unit in space. 
2. The outcomes must occur at random.
   • In an experiment or situation all the outcomes must occur randomly. 
   • For example, the time you must wait at an ATM is occurs randomly. Time is a unit of measurement and is continuous.
3. The outcomes must vary.
   • There needs to be more than one outcome and thus the outcomes vary.
   • For a CRV, the outcomes will vary because they will always exist over an interval. In fact, when we're observing data or an experiment that is measured, we cannot examine one single outcome. This will be explained later on when we delve into CRVs in more detail.

A DRV vs a CRV

When determining whether you're looking at a DRV or a CRV, you firstly need to make sure that the experiment or situation is random and varies.

You then need to consider whether the outcomes consist of discrete or continuous values. 

Remember that what you're asking yourself is whether the outcomes are counted or measured.

Worked Example

The mass of each egg in an $18$18-pack carton of Extra Large Eggs ranges in weight from $67$67 g to $72$72 g.

(a)  Can the weight of the eggs in the carton be modelled by a probability distribution?

Think: We need to check firstly that the outcomes (the weights of the eggs) are random and vary. We then need to think about what sort of data we're dealing with.

Do: The weights certainly vary because we're told they range between $67$67 g and $72$72 g. We know the eggs are randomly assigned to each space in the egg carton and we'd choose one at random.

We also know that we'd need to measure the weight of each egg, not count it. 

Therefore we can say that this situation is modelled by a continuous random variable.

(b)  Which of the following graphs best models the shape of this continuous probability distribution?

Think: Because we don't have a lot of experience with CRVs, we'll have to really think about what we expect to happen in this real-life situation. You have to use some common sense!

Do: Let's take a look at what each graph is telling us.

• The first graph, in the top left, is showing that the probability of each outcome is uniform.
• The second graph, in the top right, is showing us that the probability at the lower end of the range of outcomes is much higher than the probability at the lower end of the range of outcomes.
• The third graph, in the bottom left, is showing us that probability of the range of outcomes is positively skewed.
• The fourth graph, in the bottom right, is showing us that the probability of the range of outcomes is symmetrical and the highest probability occurs at the middle of the range.

When we think about the weights of the eggs in a carton that is advertised as containing Extra Large Eggs, we should expect that the majority of the eggs will be the same size and weight and that there's a lower chance that an egg will be at the lower or higher range of weights. So the mean weight should have the highest probability. We are therefore looking at the fourth graph as our answer.