Random Variables

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.

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`.

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

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

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.

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.

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:

- All outcomes are numerical and discrete - this means we can't have categorical or continuous numerical data.
- 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. - 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$0≤`p`≤1.

Is the following a probability distribution?

$x$x |
$2$2 | $4$4 | $6$6 | $8$8 |
---|---|---|---|---|

$p\left(x\right)$p(x) |
$0.2$0.2 | $0.4$0.4 | $0.6$0.6 | $0.8$0.8 |

No

AYes

BNo

AYes

B

Consider the following table.

$x$x |
$2$2 | $4$4 | $5$5 | $6$6 | $7$7 |
---|---|---|---|---|---|

$P$P$($($X=x$X=x$)$) |
$0.1$0.1 | $0.25$0.25 | $0.3$0.3 | $0.15$0.15 | $0.2$0.2 |

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

Select all that apply.

$\Sigma P\left(x\right)=1$Σ

`P`(`x`)=1A$X$

`X`is discrete numericalB$0\le P$0≤

`P`$\left(x\right)\le1$(`x`)≤1C$\Sigma P\left(x\right)=1$Σ

`P`(`x`)=1A$X$

`X`is discrete numericalB$0\le P$0≤

`P`$\left(x\right)\le1$(`x`)≤1CTherefore, does this table represent a discrete probability distribution?

No

AYes

BNo

AYes

B

Consider the following graph.

Loading Graph...

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

Select all that apply.

$0\le P$0≤

`P`$\left(x\right)\le1$(`x`)≤1A$\Sigma P\left(x\right)=1$Σ

`P`(`x`)=1B$X$

`X`is discrete numericalC$0\le P$0≤

`P`$\left(x\right)\le1$(`x`)≤1A$\Sigma P\left(x\right)=1$Σ

`P`(`x`)=1B$X$

`X`is discrete numericalCTherefore, does this graph represent a discrete probability distribution?

Yes

ANo

BYes

ANo

B

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

Apply probability distributions in solving problems