Random Variables Objectives At the end of this lesson you should be able to 1 2 3 4 5 Define a random variable Identify whether a random variable is discrete or continuous Create an outcome space related to a probability model Create a probability mass function and its related probability histogram Calculate probabilities given either a probability mass function or its related probability histogram Background Let s establish a little vocabulary If a process such as a function creates a set of non continuous results or values it is said to produce discrete values or to be a discrete function The collection of all such results are our outcome space in probability Example 1 Flip a coin The discrete results are heads tails We can assign a numerical to these two results While it might seem prejudicial we could let H 1 and T 0 Then the discrete values would be 0 1 2 Roll a single die The discrete values are 1 2 3 4 5 6 Nothing else can be produced from the die If we wish to make this a numerical result the point total on the die does the job nicely 3 Roll a pair of dice once While we might wish to list the ordered pairs numerical work with a pair of dice usually centers on the point total sometimes called the point achieved by the roll Then the discrete values are 2 3 4 11 12 Another way to do it would be to number the pairs Say assign 1 1 1 through 6 6 36 4 Birth a baby The results are boy girl We can use the same kind of numerical assignment as in Example 1 However I am not stupid enough to choose which gets the zero 5 Select a shoe size The practical values begin at 0 really and are numbered in halves The largest men s size I have heard of is a U S size 37 To complicate the issue we also have men s women s and children s size scales which are different around the world However were we to take all the common sizes and assign their length in centimeters it is still a discrete set 6 Select a specified number of elements from some set of fixed size as either a permutation or a combination Regardless of method the number of ways to do the selection is a whole number The outcome space is also bounded Not all probability functions are discrete So far we have only dealt with that type However let s get a definition now for a continuous model Arizona State University Department of Mathematics and Statistics Random Variables If a process or function produces a set of continuous values it is said to be continuous or a continuous function The collection of all such results are still our outcome space in probability Example 1 Pick a time from a infinitely accurate digital stopwatch With this unrealistic condition we can achieve any time reading from zero to the life of the clock s battery While the outcome space is bounded it is still continuous Make the stopwatch solar powered and we might have to wait until it fails or the sun stops shining 2 Pick a point on the number line on the open interval 0 1 This is definitely a bounded continuous space 3 Pick a point on the number line Not only is it continuous and unbounded but we have no more choices than in Example 2 4 Choose a mortgage interest rate Aside from our rather silly assumption that they should always be nice mixed numbers between 0 and 100 a computer would probably be unperturbed by an interest rate of 2 100 I would love a rate less than 1 42 5 We could haggle about actual foot size not shoe size This could be a continuous outcome space since any size between zero and perhaps that bizarre 37 might be measured with a sufficiently accurate ruler Notice that it doesn t matter if the outcome space is bounded or not The only defining property for a discrete outcome space is that to move among the outcomes we have to skip along them For a continuous outcome space we can move smoothly along the outcomes just as for any continuous function Random Variable Defined So what is a random variable anyway We answered it with the last pithy statement A random variable is nothing more than the function we use to create a numerical outcome space even if the process we use creates something like heads or tails Examples 1 Roll a single die The discrete results are in S 1 2 3 4 5 6 Let X represent the function random variable so that for any possible outcome s S 1 2 3 4 5 6 X s s This particular random variable is the identity function restricted to the first six whole number It assigns the six possibilities as the numerical outcomes Its associated outcome space is what we have previously called the range in working with functions 2 Roll a pair of dice once Let X be the random variable such that X a b a b for a 1 2 3 4 5 6 and b 1 2 3 4 5 6 Then the outcome space is S 2 3 4 11 12 2 of 6 Arizona State University Department of Mathematics and Statistics Random Variables 3 Throw a dart at a rectangular board looking like the one to the right Then let X represent the function that assigns s S 1 0 1 3 4 2 13 23 by X s s Again notice the identity function is doing the work as the random variable 4 Throw a dart at a rectangular board looking like the one to the right Now let X represent the piecewise function that assigns a numerical value to 1 3 1 4 2 23 0 13 s S 1 0 1 3 4 2 13 23 by this rule 2for s 0 1 13 23 X s 0for s 1 2 3 in dollars The outcome space is v V 2 0 2 2for s 4 So what just happened I used V to name the space because I was thinking of value to the player The random variable function just assigned a win of 2 break even 0 value or loss of 2 to throwing at the dartboard That s all the random variable does It is random because just as with the dice dart board and many other conceptually similar situations we cannot state with certainty what specific value will appear When such randomness doesn t exist we could say the game is fixed Probability Mass Functions While we are focused on the discrete situations lets talk about the probability mass function a k a probability function frequency function relative frequency function probability density function pdf For short we will use pmf as the authors of your textbook do The second most popular name pdf will be used in a different context The pmf is nothing more than a probability with an attitude We define …