Discrete random variables

There are two types of random variables, discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring. For example, let X = temperature of a randomly selected day in June in a city. The value of X can be 68°, 71.5°, 80.6°, or 90.32°. These values are obtained by measuring by a thermometer. Another example of a continuous random variable is the height of a randomly selected high school student. The value of this random variable can be 5'2", 6'1", or 5'8". Those values are obtained by measuring by a ruler.

A discrete probability distribution function has two characteristics:

1. Each probability is between zero and one, inclusive.
2. The sum of the probabilities is one.

Condensed — the full section is in OpenStax Statistics.

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution:

Let X = the number of years a new hire will stay with the company.

Let P(x) = the probability that a new hire will stay with the company x years.

Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.

Use the following information to answer the next two exercises: Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random.

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time.

The expected value of a discrete random variable X, symbolized as E(X), is often referred to as the long-term average or mean (symbolized as μ). This means that over the long term of doing an experiment over and over, you would expect this average. For example, let X = the number of heads you get when you toss three fair coins. If you repeat this experiment (toss three fair coins) a large number of times, the expected value of X is the number of heads you expect to get for each three tosses on average.

The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens.

Condensed — the full section is in OpenStax Statistics.

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution:

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:

• Let X = the number of years a student will study ballet with the teacher.
• Let P(x) = the probability that a student will study ballet x years.

There are three characteristics of a binomial experiment:

1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
2. There are only two possible outcomes, called success and failure, for each trial. The outcome that we are measuring is defined as a success, while the other outcome is defined as a failure. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1.
3. The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same. Let us look at several examples of a binomial experiment.
   Example 1: Toss a fair coin once and record the result.
   This is a binomial experiment since it meets all three characteristics. The number of trials n = 1. There are only two outcomes, a head or a tail, of each trial. We can define a head as a success if we are measuring number of heads. For a fair coin, the probabilities of getting head or tail are both .5. So, p = q − .5. Both p and q remain the same from trial to trial. This experiment is also called a Bernoulli trial, named after Jacob Bernoulli who, in the late 1600s, studied such trials extensively. Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli trial. A binomial experiment takes place when the number of successes is counted in one or more Bernoulli trials.
   Example 2: Randomly guess a multiple choice question has A, B, C and D four options.
   This is a binomial experiment since it meets all three characteristics. The number of trials n = 1. There are only two outcomes, guess correctly or guess wrong, of each trial. We can define guess correctly as a success. For a random guess (you have no clue at all), the probability of guessing correct should be \(\frac{1}{4}\) because there are four options and only one option is correct. So, and \(p=\frac{1}{4}\) and \(q=1-p=1-\frac{1}{4}=\frac{3}{4}\). Both p and q remain the same from trial to trial. This experiment is also a Bernoulli trial. It meets the characteristics two and three and n = 1.
   Example 3: Toss a fair coin five times and record the result.
   This is a binomial experiment since it meets all three characteristics. The number of trials n = 5. There are only two outcomes, head or tail, of each trial. If we define head as a success, then p = q = 0.5. Both p and q remain the same for each trial. Since n = 5, this experiment is not a Bernoulli trial although it meets the characteristics two and three.
   Example 4: Randomly guess 10 multiple choice questions in an exam. Each question has A, B, C and D four options.
   This is a binomial experiment since it meets all three characteristics. The number of trials n = 10. There are only two outcomes, guess correctly or guess wrong, of each trial. We can define guess correctly as a success. As we explained in example 2, \(p=\frac{1}{4}\) and \(q=1-p=1-\frac{1}{4}=\frac{3}{4}\). Both p and q remain the same for each guess. Since n = 10, this experiment is not a Bernoulli trial.
   The next two experiments are not binomial experiments.
   Example 5: Randomly select two balls from a jar with five red balls and five blue balls without replacement. This means we select the first ball, and then without returning the selected ball into the jar, we will select the second ball.
   This is not a binomial experiment since the third characteristic is not met. The number of trials n = 2. There are only two outcomes, a red ball or a blue ball, of each trial. If we define selecting a red ball as a success, then selecting a blue ball is a failure. The probability of getting the first ball red is \(\frac{5}{10}\) since there are five red balls out of 10 balls. So, \(p=\frac{5}{10}\) and \(q=1-p=1-\frac{5}{10}=\frac{5}{10}\). However, p and q do not remain the same for the second trial. If the first ball selected is red, then the probability of getting the second ball red is \(\frac{4}{9}\) since there are only four red balls out of nine balls. But if the first ball selected is blue, then the probability of getting the second ball red is \(\frac{5}{9}\) since there are still five red balls out of nine balls.
   Example 6: Toss a fair coin until a head appears.
   This is not a binomial experiment since the first characteristic is not met. The number of trials n is not fixed. n could be 1 if a head appears from the first toss. n could be 2 if the first toss is a tail and the second toss is a head. So on and so forth.
   More examples of binomial and non-binomial experiments will be discussed in this section later.

Condensed — the full section is in OpenStax Statistics.

X ~ B(n, p)

Read this as X is a random variable with a binomial distribution. The parameters are n and p: n = number of trials, p = probability of a success on each trial.

Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed.

Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

There are three main characteristics of a geometric experiment:

1. Repeating independent Bernoulli trials until a success is obtained. Recall that a Bernoulli trial is a binomial experiment with number of trials n = 1. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bull's-eye until you hit the bull's-eye. The first time you hit the bull's-eye is a success so you stop throwing the dart. It might take six tries until you hit the bull's-eye. You can think of the trials as failure, failure, failure, failure, failure, success, stop.
2. In theory, the number of trials could go on forever. There must be at least one trial.
3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial. p + q = 1 and q = 1 − p. For example, the probability of rolling a three when you throw one fair die is \(\frac{1}{6}\). This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is q = \(\frac{\text{5}}{\text{6}}\), the probability of a failure. The probability of getting a three on the fifth roll is \((\frac{5}{6})(\frac{5}{6})(\frac{5}{6})(\frac{5}{6})(\frac{1}{6})\) = .0804.

X = the number of independent trials until the first success.

p = the probability of a success, q = 1 – p = the probability of a failure.

There are shortcut formulas for calculating mean μ, variance σ², and standard deviation σ of a geometric probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book.

Condensed — the full section is in OpenStax Statistics.

X ~ G(p)

Read this as X is a random variable with a geometric distribution. The parameter is p; p = the probability of a success for each trial.

Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask.

There are five characteristics of a hypergeometric experiment:

1. You take samples from two groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players.
4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is \(\frac{13}{24}\). The probability of picking a man second is \(\frac{11}{23}\) if a woman was picked first. It is \(\frac{10}{23}\) if a man was picked first. The probability of the second pick depends on what happened in the first pick.
5. You are not dealing with Bernoulli trials.

The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest.

Condensed — the full section is in OpenStax Statistics.

X ~ H(r, b, n)

Read this as X is a random variable with a hypergeometric distribution. The parameters are r, b, and n: r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample.

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

There are two main characteristics of a Poisson experiment.

1. The Poisson probability distribution gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages.
2. The Poisson distribution may be used to approximate the binomial if the probability of success is small (such as .01) and the number of trials is large (such as 1,000). You will verify the relationship in the homework exercises. n is the number of trials, and p is the probability of a success.

The random variable X = the number of occurrences in the interval of interest.

X ~ P(μ)

Read this as X is a random variable with a Poisson distribution. The parameter is μ (or λ); μ (or λ) = the mean for the interval of interest.

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.