5.1: Random Variables
Learning Objectives
- Distinguish between discrete and continuous random variables
- Find the probability distribution of discrete random variables, and use it to find the probability of events of interest.
- Find the mean and variance of a discrete random variable, and apply these concepts to solve real-world problems.
- Apply the rules of means and variances to find the mean and variance of a linear transformation of a random variable and the sum of two independent random variables.
- Fit the binomial model when appropriate, and use it to perform simple calculations.
- Explain how a density function is used to find probabilities involving continuous random variables.
- Find probabilities associated with the normal distribution.
- Use the normal distribution as an approximation of the binomial distribution, when appropriate.
In the previous two parts we’ve learned principles and tools that help us find probabilities of events in general. Now that we’ve become proficient at doing that, we’lltalk about random variables. Just like any other variable, random variables can take on multiple values. What differentiates random variables from other variables is that the values for these variables are determined by a random trial, random sample, or simulation. The probabilities for the values can be determined by theoretical or observational means. Such probabilities play a vital role in the theory behind statistical inference, our ultimate goal in this course.
Introduction
We first discussed variables in the Exploratory Data Analysis portion of the course. A variable is a characteristic of an individual. We also made an important distinction between categorical variables, whose values are groups or categories (and an individual can be placed into one of them), and quantitative variables, which have numerical values for which arithmetic operations make sense. In the previous two modules, we focused mostly on events which arise when there is a categorical variable in the background: blood type, pierced ears (yes/no), gender, on time delivery (yes/no), side effect (yes/no), etc. Now we will begin to consider quantitative variables that arise when a random experiment is performed. We will need to define this new type of variable.
- random variable
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A random variable assigns a unique numerical value to the outcome of a random experiment.
A random variable can be thought of as a function that associates exactly one of the possible numerical outcomes to each trial of a random experiment. However, that number can be the same for many of the trials.
Before we go any further, here are some simple examples:
Example
Theoretical
Consider the random experiment of flipping a coin twice. The sample space of possible outcomes is S = { HH, HT, TH, TT }.
Now, let’s define the variable X to be the number of tails that the random experiment will produce.
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If the outcome is HH, we have no tails, so the value for X is 0.
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If the outcome is HT, we got one tail, so the value for X is 1.
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If the outcome is TH, we again got one tail, so the value for X is 1.
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Lastly, if the outcome is TT, we got two tails, so the value for X is 2.
As the definition suggests, X is a quantitative variable that takes the possible values of 0, 1, or 2.
It is random because we do not know which of the three values the variable will eventually take. We can ask questions like:
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What is the probability that X will be 2? In other words, what is the probability of getting 2 tails?
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What is the probability that X will be at least 1? In other words, what is the probability of getting at least 1 tail?
As you can see, random variables are not really a new thing, but just a different way to look at the same problem.
Note that if we had tossed a coin three times, the possible values for the number of tails would be 0, 1, 2, or 3. In general, if we toss a coin “n” times, the possible number of tails would be 0, 1, 2, 3, … , or n.
Example
Observational
Consider getting data from a random sample on the number of ears in which a person wears one or more earrings.
We define the variable X to be the number of earsin which a randomly selected person wears an earring.
If the selected person does not wear any earrings, then X = 0.
If the selected person wears earrings in either the left or the right ear, then X = 1.
If the selected person wears earrings in both ears, then X = 2.
As the definition suggests, X is a quantitative variable which takes the possible values of 0, 1, or 2. We can ask questions like:
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What is the probability that a randomly selected person will have earrings in both ears?
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What is the probability that a randomly selected person will not be wearing any earrings in either ear?
Note…We identified the first example as theoretical and the second as observational. Let’s discuss the distinction.
To answer probability questions about a theoretical situation, we only need the principles of probability. However, if we have an observational situation, the only way to answer probability questions is to use the relative frequency we obtain from a random sample.
Here is a different type of example:
Example
Lightweight Boxer
Assume we choose a lightweight male boxer at random and record his exact weight. According to the boxing rules, a lightweight male boxer must weigh between 130 and 135 pounds, so the sample space here is S = { All the numbers in the interval 130-135 }. (Note that we can’t list all the possible outcomes here.)
We’ll define X to be the weight of the boxer (again, as the definition suggests, X is a quantitative variable whose value is the result of our random experiment). Here X can take any value between 130 and 135. We can ask questions like:
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What is the probability that X will be more than 132? In other words, what is the probability that the boxer will weigh more than 132 pounds?
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What is the probability that X will be between 131 and 133? In other words, what is the probability that the boxer weighs between 131 and 133 pounds?
What is the difference between the random variables in these examples? Let’s see:
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They all arise from a random experiment (tossing a coin twice, choosing a person at random, choosing a lightweight boxer at random).
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They are all quantitative (number of tails, number of ears, weight).
Where they differ is in the type of possible values they can take: In the first two examples, X has three distinct possible values: 0, 1, and 2. You can list them. In contrast, in the third example, X takes any value in the interval 130-135, and thus the possible values of X cover an infinite range of possibilities, and cannot be listed.
A random variable like the one in the first two examples, whose possible values are a list of distinct values, is called a discrete random variable. A random variable like the one in the third example, that can take any value in an interval, is called a continuous random variable.
Just as the distinction between categorical and quantitative variables was important in Exploratory Data Analysis, the distinction between discrete and continuous random variables is important here, as each one gets a different treatment when it comes to calculating probabilities and other quantities of interest.
Before we go any further, a few observations about the nature of discrete and continuous random variables should be mentioned.
Comments
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Sometimes, continuous random variables are “rounded” and are therefore “in a discrete disguise.”
For example:
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time spent watching TV in a week, rounded to the nearest hour (or minute)
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outside temperature, to the nearest degree
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a person’s weight, to the nearest pound.
Even though they “look like” discrete variables, these are still continuous random variables, and we will in most cases treat them as such.
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On the other hand, there are some variables which are discrete in nature, but take so many distinct possible values that it will be much easier to treat them as continuous rather than discrete.
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the IQ of a randomly chosen person
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the SAT score of a randomly chosen student
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the annual salary of a randomly chosen CEO, whether rounded to the nearest dollar or the nearest cent
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Sometimes we have a discrete random variable but do not know the extent of its possible values. For example: How many accidents will occur in a particular intersection this month? We may know from previously collected data that this number is from 0-5. But, 6, 7, or more accidents could be possible.
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A good rule of thumb is that discrete random variables are things we count, while continuous random variables are things we measure.
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We counted the number of tails and the number of ears with earrings. These were discrete random variables.
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We measured the weight of the lightweight boxer. This was a continuous random variable.
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Often we can have a subject matter for which we can collect data that could involve a discrete or a continuous random variable, depending on the information we wish to know.
Example
Soft Drinks
Suppose we want to know how many days per week you drink a soft drink. The sample space would be S = { 0, 1, 2, 3, 4, 5, 6, 7 }. There are a finite number of values for this variable. This would be a discrete random variable.
Instead, suppose we want to know how many ounces of soft drinks you consume per week. Even if we round to the nearest ounce, the answer is a measurement. Thus, this would be a continuous random variable.
Example
x-bar
Suppose we are interested in the weights of all males. We take a random sample and get the mean for that sample, namely ¯x. We then take another random sample (with the same sample size) and get another ¯x.
We would expect the values of the ¯xs from these two samples to be different, but pretty close in value.
Each time we take a sample we’ll get a different ¯x. We will take lots of samples and thus get many ¯xs.
The value of ¯x from these repeated samples is a random variable. Since it can take on any value within an interval of possible male weights it is a continuous random variable.
Did I get this?
Choose a college student at random. Decide whether each of the following is a discrete or continuous random variable:
We devote a great deal of attention to random variables, since random variables and the probabilities that are associated with them play a vital role in the theory behind statistical inference, our ultimate goal in this course.
This chapter is organized in two parts; one on discrete random variables, and one on continuous. We’ll start with discrete.
