Chapter 4 Discrete Random Variables

OPENSTAX STATISTICS

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Objectives

• 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.
• Classify discrete word problems by their distributions.

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Section 4.1

PROBABILITY DISTRIBUTION FUNCTION (PDF) FOR A DISCRETE

RANDOM VARIABLE

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Random Variable Notation

• 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.
• Upper case letters such as 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.

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Probability Distribution Function �for a Discrete Random Variable

• 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.

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Probability Distribution Function �for a Discrete Random Variable Example

• 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 per week a newborn baby's crying 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 takes on the values 0, 1, 2, 3, 4, 5.
• This is a discrete PDF because:
• Each P(x) is between zero and one, inclusive.
• The sum of the probabilities is one.

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Example

• Suppose Nancy has classes three days a week. She attends classes three days a week 80% of the time, two days 15% of the time, one day 4% of the time, and no days 1% of the time. Suppose one week is randomly selected.
• a. Let X = the number of days Nancy ____________________.
• b. X takes on what values?
• c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table). The table should have two columns labeled x and P(x). What does the P(x) column sum to?

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Example - Answers

• a. Let X = the number of days Nancy attends class per week.
• b. 0, 1, 2, and 3
• c. See table:

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Section 4.2

MEAN OR EXPECTED VALUE AND STANDARD DEVIATION

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Mean or Expected Value

• 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 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).

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Mean or Expected Value (Continued)

• When evaluating the long-term results of statistical experiments, we often want to know the “average” outcome.
• This “long-term average” is known as the mean or expected value of the experiment and is denoted by the Greek letter μ.
• In other words, after conducting many trials of an experiment, you would expect this average value.
• NOTE: 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.

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Formulas for Expected Value and Standard Deviation

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Why Do We Have Different Formulas?

One Situation, Two Ways

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Expected Value Example

• A men's soccer team plays soccer zero, one, or two days a week. The probability that they play zero days is 0.2, the probability that they play one day is 0.5, and the probability that they play two days is 0.3. Find the long-term average or expected value, μ, of the number of 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 x*P(x). In this column, you will multiply each x value by its probability.
• Add the last column x*P(x) 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.

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Example

• Suppose these are the probabilities for each outcome. Find the missing value:

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a. What do the probabilities need to sum to?

b. P(x = 4) = _______

c. P(x ≥ 5) = _______

d. On average, how long would you expect a new hire to stay with the company? (see Excel workbook)

e. What is the standard deviation?

Example - Answers

• Suppose these are the probabilities for each outcome. Find the missing value:

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a. They should sum to 1. The missing value is 0.1

b. P(x = 4) = 0.1

c. P(x ≥ 5) = 0.15

d. 2.43

e. 1.65

Example

• Find the expected value and standard deviation from the expected value table. See the Excel workbook.

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Example - Answers

• Expected value should be 5.4 and standard deviation should be 1.8.

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Example

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Example - Answers

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Example

• Toss a fair, six-sided die twice. Let X = the number of faces that show an even number. Construct a table of outcomes and calculate the mean μ and standard deviation σ of X. Create your own unique Excel spreadsheet.

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Example - Answers

• Tossing one fair six-sided die twice has the same sample space as tossing two fair six-sided dice. The sample space has 36 outcomes, which can be used to create a probability table (next slide).

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Example - Answers

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Section 4.3

THE BINOMIAL DISTRIBUTION

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Binomial Distribution

• 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 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.

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Example of “Success”

• At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term.
• A "success" could be defined as an individual who withdrew.
• The random variable X = the number of students who withdraw from the randomly selected elementary physics class.

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Notation for the Binomial Probability Distribution Function

• The outcomes of a binomial experiment fit a binomial probability distribution.
• The random variable X = the number of successes obtained in the n independent trials.
• 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.
• Binomial probabilities are found by using the binomial distribution function. Stating the probability question mathematically is the start.

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Example of Binomial Distribution Problem

• Suppose you play a game that you can only either win or lose. The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define X as the number of wins, then X takes on the values 0, 1, 2, 3, ..., 20. The probability of a success is p = 0.55. The probability of a failure is q = 0.45. The number of trials is n = 20. The probability question can be stated mathematically as P(x = 15).

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The Binomial Formula

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where b(x) is the probability of X successes in n trials when the probability of a success in ANY ONE TRIAL is p. And q=(1-p) and is the probability of a failure in any one trial

• This is the formula that tells the number of unique unordered subsets of size x that can be created from n unique elements.
• The formula is read “n combinatorial x”.
• Sometimes it is read as “n choose x."
• We can use Excel to calculate this for us (combin function) or we can compute it using a calculator

Other Binomial Distribution Formulas

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Example

• A coin has been altered to weight the heads outcome from 0.5 to 0.25 and is flipped 5 times. Each flip is independent.
• Let X = the number of heads in 5 flips of the coin. X takes on the values 0, 1, 2, 3, 4, 5.

Since the coin is altered to result in p = 0.25, q is 0.75. The number of trials is n = 5.

a. What is the probability of getting more than 3 heads?

b. What is the probability of getting exactly 2 heads?

c. What is the expected number of heads in 5 flips?

• We will need to create a table of probabilities, refer to the filled in Excel spreadsheet.

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Example - Answers

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Example

• In the 2013 Jerry’s Artarama art supplies catalog, there are 560 pages. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists. Use the Excel spreadsheet to build out this situation.
• What values does x take on?
• What is the probability distribution? Find the following probabilities:
1. the probability that two pages feature signature artists
2. the probability that at most six pages feature signature artists
3. the probability that more than three pages feature signature artists.
• Using the formulas, calculate the (i) mean and (ii) standard deviation.

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Example - Answers

1. x = 0, 1, 2, 3, 4, 5, 6, 7, 8
2. X ~ B (100,8560)
1. P(x = 2) = = 0.2466
2. P(x ≤ 6) = = 0.9994
3. P(x > 3) = 1 – P(x ≤ 3) = = 1 – 0.9443 = 0.0557
3. i) Mean = 1.4286 ii) Standard Deviation = 1.1867

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Example

• According to a recent article the average number of babies born with significant hearing loss (deafness) is approximately two per 1,000 babies in a healthy baby nursery. The number climbs to an average of 30 per 1,000 babies in an intensive care nursery.
• Suppose that 1,000 babies from healthy baby nurseries were randomly surveyed. Find the probability that exactly two babies were born deaf.
• Create your own Excel model for this situation.

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Example - Answer

• According to a recent article the average number of babies born with significant hearing loss (deafness) is approximately two per 1,000 babies in a healthy baby nursery. The number climbs to an average of 30 per 1,000 babies in an intensive care nursery.
• Suppose that 1,000 babies from healthy baby nurseries were randomly surveyed. Find the probability that exactly two babies were born deaf.
• Answer: 0.2709

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Other Discrete Probability Distributions Exist!

• Though we are not covering them, please note that other discrete distributions exist, such as:
• Geometric Distribution
• Poisson Distribution
• Hypergeometric Distribution

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