Discrete Random Variable

When thinking about what a Discrete Random Variable (or DRV for short) actually is, the name itself tells you most about what you need to know.

But before we get into explaining each of the words (discrete, random and variable), we'll start with an example which will make it very clear what we're talking about when we say DRV.

A cat with three kittens

Let's say there's a cat who's about to give birth to three kittens. Before they are born we know each will either be male or female. The exact combination of male and female kittens is unknown before they are born. We can instead consider all possible outcomes of the birth. An easy way to do this is to represent the possibilities with the following tree diagram.

What we've done so far is create a simple sample space. You've done this heaps of times before. What we need to do now though, is choose something to focus on in this situation. There's only two things to focus on in this situation: either we focus on the number of female kittens or the number of male kittens.

Let's focus on the number of female kittens. 

How many female kittens will we see once all the kittens are born? Either $0,1,2$0,1,2 or $3$3.

So the number of female kittens will vary and will occur at random.

When a quantity varies, we can define a variable. In this case let's define $X$X as the number of female kittens born.

$x$x will have values of $0,1,2$0,1,2 and $3$3, where $x$x represents the possible outcomes for event $X$X occurring.

Each of these values for $X$X have a particular chance or probability of occurring. We can use our tree diagram and a table to summarise these probabilities.

$x$x	$0$0	$1$1	$2$2	$3$3
$P(X=x)$P(X=x)	$\frac{1}{8}$18​	$\frac{3}{8}$38​	$\frac{3}{8}$38​	$\frac{1}{8}$18​

What we've created here is a discrete probability distribution and represented it with an individual probability table.

$x$x represents the individual outcomes of event $X$X occurring.

$P(X=x)$P(X=x) represents the probability that outcome $x$x occurs for random variable $X$X. Put more simply, it's the probability of each of the outcomes occurring.

We could also represent this information with a cumulative probability table. That is, $P\left(X\le x\right)$P(X≤x) represents the probability of the outcome being less than or equal to $x$x.

$x$x	$0$0	$1$1	$2$2	$3$3
$P\left(X\le x\right)$P(X≤x)	$\frac{1}{8}$18​	$\frac{4}{8}$48​	$\frac{7}{8}$78​	$\frac{8}{8}$88​

Defining a Discrete Random Variable

Now that we've taken a look at an example, let's use it to really understand what a DRV is. 

1. Firstly, the outcomes of the situation or the experiment must take discrete values.
   • Remember that discrete data is numerical and takes on integer values. You'll know your're talking about a DRV when you have counted the number of possible outcomes.
   • In our example, we counted that there could be either $0,1,2$0,1,2 or $3$3 female kittens.
2. The outcomes must occur at random.
   • In an experiment or situation all the outcomes must occur randomly. 
   • In our example, we didn't know how many female kittens would be born and it happens at random.
3. The outcomes must vary.
   • There needs to be more than one outcome and thus the outcomes vary.
   • If in our example we'd only been interested in the particular situation where $2$2 female kittens were born, we'd only have one outcome: $2$2 females and $1$1 male. So this wouldn't then constitute what's needed to be a DRV.

 

Worked Examples

Question 1

question 2