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.
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.
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:
Is the following a probability distribution?
Consider the following table.
Identify which conditions for a discrete probability distribution are evident in the table.
Select all that apply.
Therefore, does this table represent a discrete probability distribution?
Consider the following graph.
Identify which conditions for a discrete probability distribution are evident in the graph.
Select all that apply.
Therefore, does this graph represent a discrete probability distribution?
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