A random variable assigns a value to the outcomes in a random situation. Random variables can be continuous, meaning that they can take on any value or they can be discrete. Discrete random variables can only take on values from a countable list.
Consider attending a baseball game at Shea Stadium, which is located next to LaGuardia Airport in New York. At a ballgame there are many random variables that may effect you while you watch the game. The temperature is considered a continuous random variable
because it can take on any value. Although we usually round to the nearest degree, it is still a continuous variable. On the other hand, the number of planes that fly over during the game is a discrete random variable. There can only be 0, 1, 2, 3, 4, etc planes that fly over. There is a countable list of numbers that the number of planes comes from. In other words, there cannot be 3.7 planes that fly over. The number of people who attend the baseball game is another example of a discrete random variable. The length
of time you wait in the security line
is another example of a continuous random variable.
Once we have identified random variables and the type of variable
it is, there are usually probabilities assigned for each possible value.
Let’s look at a discrete case in which you toss a coin two times. What are the possible outcomes and probabilities?
First, the possible outcomes.
HH, HT, TH, and TT
In other words, there can be 0, 1, or 2 tails:
Most of the time this information is organized in table
form and called a discrete probability distribution
. It is describing a discrete random variable
and it shows how the probabilities are distributed to all the outcomes. Hence the name - discrete probability
distribution. For our example, it would look like
Note that all the probabilities on the “probability” line
of the distribution must all add up to 1.
There are a variety of problems that you can be asked to solve based on a discrete probability
distribution. You could be asked to do something as simple as complete a table
or perform more complicated tasks such as finding the mean
and standard deviation of a distribution. Also there are special kinds of discrete probability
distributions, one of which is a binomial
distribution. Examining these types of problems can take up an entire chapter in a probability/statistics book and are too numerous to explore in a single lesson. For additional information, ask your teacher for help or to direct you to an appropriate statistics textbook.
As discrete random variables can have distribution functions, there is a similar situation for continuous random variables. However, since each and every possibility cannot be listed for continuous variables as is the case with discrete random variables, we have a slightly different look and terminology.
When dealing with a continuous random variable
and the assigned probabilities, we refer to a probability density function
and deal with the area under a curve
Suppose the shuttle service from the airport parking lot to the terminal arrives at the parking lot every 10 minutes. If you arrive at a random time, how long should you expect to wait? That is dependent on the density
function, in this case a uniform density
function. Another very common type of probability density function
is the normal distribution. As with discrete probability
distribution function, there are entire chapters in books devoted to this topic and too numerous to discuss in one, or even ten, lessons.