New Zealand
Level 8 - NCEA Level 3

# Discrete Random Probability Distribution

Lesson

As we have seen, a Discrete Random Variable, or DRV for short, is when each outcome in a random experiment is assigned a number.

A Continuous Random Variable, or CRV for short, is when we examine the outcomes over an interval for a random experiment.

An easy way to think about it is this: we count the outcomes for a DRV and we measure the outcomes for a CRV.

Now that we know the difference between a DRV and CRV, we want to take a closer look at DRVs, and in particular, we want to look at the distribution of the probabilities.

## The Probability Distribution of a Discrete Random Variable

For a DRV, each outcome is assigned a probability.

For example, let's say we toss a coin twice and we're interested in how many tails we see.

Firstly we can define our DRV. Let $X$X be the number of tails in a two coin toss.

We then know that $X$X can take on the values of $0$0, $1$1 or$2$2.

The probability distribution for X will show us the probabilities for each of these outcomes. An easy way to do that for this example is to look at a tree diagram.

We'll now examine our tree diagram and tabulate the probabilities.

$x$x 0 1 2
$P(X=x)$P(X=x) $\frac{1}{4}$14 $\frac{2}{4}$24 $\frac{1}{4}$14

What we've just created is the probability distribution for the random variable $X$X.

## How can we represent a probability distribution for a DRV?

There are a few ways we represent a probability distribution for a DRV.

1. We can create a table of values. This is the most common method.
2. We can create a function. You will see this very soon.
3. We can draw a graph.

## The graph of a probability distribution for a DRV.

The graph of a probability distribution of a DRV is a probability histogram.

Either of these two graphs can be used to represent the DRV X that we were discussing earlier.

## Uniform vs Non-Uniform DRVs

A uniform DRV is when all outcomes of the experiment are equally likely.

For example, when you roll a dice once, the probability of rolling a $1$1,$2$2, $3$3, $4$4,$5$5 or $6$6 are all equally likely, or uniform.

The probability histogram below of one roll of the dice shows this uniform shape nicely.

A non-uniform DRV is when not all outcomes are equally likely. This is the vast majority of DRVs that we'll see.

## The Properties of a Probability Distribution of a DRV

If we examine what appears to be a probability distribution, either in a table, a graph or a function, we need to make sure each of the following three properties apply:

1. All outcomes are numerical and discrete - this means we can't have categorical or continuous numerical data.
2. The probabilities of all possible outcomes add to $1$1 - this makes sense when you think about it, if they didn't all add up to $1$1 then we must be missing one of the outcomes or have one of the probabilities wrong.
3. The probability $p$p of all outcomes is greater than or equal to $0$0 and less than or equal to $1$1, that is $0\le p\le1$0p1.

#### Worked Examples

##### Question 1

Is the following a probability distribution?

 $x$x $p\left(x\right)$p(x) $2$2 $4$4 $6$6 $8$8 $0.2$0.2 $0.4$0.4 $0.6$0.6 $0.8$0.8
1. No

A

Yes

B

No

A

Yes

B

##### Question 2

Consider the following table.

 $x$x $P$P$($($X=x$X=x$)$) $2$2 $4$4 $5$5 $6$6 $7$7 $0.1$0.1 $0.25$0.25 $0.3$0.3 $0.15$0.15 $0.2$0.2
1. Identify which conditions for a discrete probability distribution are evident in the table.

Select all that apply.

$\Sigma P\left(x\right)=1$ΣP(x)=1

A

$X$X is discrete numerical

B

$0\le P$0P$\left(x\right)\le1$(x)1

C

$\Sigma P\left(x\right)=1$ΣP(x)=1

A

$X$X is discrete numerical

B

$0\le P$0P$\left(x\right)\le1$(x)1

C
2. Therefore, does this table represent a discrete probability distribution?

No

A

Yes

B

No

A

Yes

B

##### Question 3

Consider the following graph.

1. Identify which conditions for a discrete probability distribution are evident in the graph.

Select all that apply.

$0\le P$0P$\left(x\right)\le1$(x)1

A

$\Sigma P\left(x\right)=1$ΣP(x)=1

B

$X$X is discrete numerical

C

$0\le P$0P$\left(x\right)\le1$(x)1

A

$\Sigma P\left(x\right)=1$ΣP(x)=1

B

$X$X is discrete numerical

C
2. Therefore, does this graph represent a discrete probability distribution?

Yes

A

No

B

Yes

A

No

B

### Outcomes

#### S8-4

Investigate situations that involve elements of chance: A calculating probabilities of independent, combined, and conditional events B calculating and interpreting expected values and standard deviations of discrete random variables C applying distributions such as the Poisson, binomial, and normal

#### 91586

Apply probability distributions in solving problems