This module serves as the introduction to Discrete Random Variables in the Elementary Statistics textbook/collection.
By the end of this chapter, the student should be able to:
Recognize and understand discrete probability distribution functions, in general.
Calculate and interpret expected values.
Recognize the binomial probability distribution and apply it appropriately.
Recognize the Poisson probability distribution and apply it appropriately (optional).
Recognize the geometric probability distribution and apply it appropriately (optional).
Recognize the hypergeometric probability distribution and apply it appropriately (optional).
Classify discrete word problems by their distributions.
A student takes a 10 question true-false quiz. Because the student had such a busy schedule, he or she could not study and randomly guesses at each answer. What is the probability of the student passing the test with at least a 70%?
Small companies might be interested in the number of long distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long distance phone calls during the peak time?
These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A random variable describes the outcomes of a statistical experiment in words. The values of a random variable can vary with each repetition of an experiment.
In this chapter, you will study probability problems involving discrete random distributions. You will also study long-term averages associated with them.
Upper case letters like X or Y denote a random variable. Lower case letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words. and x is given as a number.
For example, let X = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT; THH; HTH; HHT; HTT; THT; TTH; HHH. Then, x = 0, 1, 2, 3. X is in words and x is a number. Notice that for this example, the x values are countable outcomes. Because you can count the possible values that X can take on and the outcomes are random (the x values 0, 1, 2, 3), X is a discrete random variable.
Toss a coin 10 times and record the number of heads. After all members of the class have completed the experiment (tossed a coin 10 times and counted the number of heads), fill in the chart using a heading like the one below. Let X = the number of heads in 10 tosses of the coin.
| x | Frequency of x | Relative Frequency of x |
|---|---|---|
Which value(s) of x occurred most frequently?
If you tossed the coin 1,000 times, what values could x take on? Which value(s) of x do you think would occur most frequently?
What does the relative frequency column sum to?
This module introduces the Probability Distribution Function (PDF) and its characteristics.
A discrete probability distribution function has two characteristics:
Each probability is between 0 and 1, inclusive.
The sum of the probabilities is 1.
A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times a newborn wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5.
P(x) = probability that X takes on a value x.
| x | P(x) |
| 0 |  |
| 1 |  |
| 2 |  |
| 3 |  |
| 4 |  |
| 5 |  |
X takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because
Each P(x) is between 0 and 1, inclusive.
The sum of the probabilities is 1, that is,
Suppose Nancy has classes 3 days a week. She attends classes 3 days a week 80% of the time, 2 days 15% of the time, 1 day 4% of the time, and no days 1% of the time. Suppose one week is randomly selected.
Let X = the number of days Nancy ____________________ .
X takes on what values?
Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in the previous example. The table should have two columns labeled x and P(x). What does the P(x) column sum to?
| x | P(x) |
| 0 | 0.01 |
| 1 | 0.04 |
| 2 | 0.15 |
| 3 | 0.80 |
This module explores the Law of Large Numbers, the phenomenon where an experiment performed many times will yield cumulative results closer and closer to the theoretical mean over time.
The expected value is often referred to as the "long-term"average or mean . This means that over the long term of doing an experiment over and over, you would expect this average.
The mean of a random variable X is μ. If we do an experiment many times (for instance, flip a fair coin, as Karl Pearson did, 24,000 times and let X = the number of heads) and record the value of X each time, the average is likely to get closer and closer to μ as we keep repeating the experiment. This is known as the Law of Large Numbers.
To find the expected value or long term average, μ, simply multiply each value of the random variable by its probability and add the products.
A men's soccer team plays soccer 0, 1, or 2 days a week. The probability that they play 0 days is 0.2, the probability that they play 1 day is 0.5, and the probability that they play 2 days is 0.3. Find the long-term average, μ, or expected value of the days per week the men's soccer team plays soccer.
To do the problem, first let the random variable X = the number of days the men's soccer team plays soccer per week. X takes on the values 0, 1, 2. Construct a PDF table, adding a column xP(x). In this column, you will multiply each x value by its probability.
Add the last column to find the long term average or expected value: (0)(0.2)+(1)(0.5)+(2)(0.3)= 0 + 0.5 + 0.6 = 1.1.
The expected value is 1.1. The men's soccer team would, on the average, expect to play soccer 1.1 days per week. The number 1.1 is the long term average or expected value if the men's soccer team plays soccer week after week after week. We say μ=1.1
Find the expected value for the example about the number of times a newborn baby's crying wakes its mother after midnight. The expected value is the expected number of times a newborn wakes its mother after midnight.
| x | P(X) | x P(X) |
|---|---|---|
| 0 |  | (0) = 0 |
| 1 |  | (1) =  |
| 2 |  | (2) =  |
| 3 |  | (3) =  |
| 4 |  | (4) =  |
| 5 |
|