5. Binomial Distribution & Discrete Random Variables

A factory produces lightbulbs in batches of 50. The probability distribution for the number of defective lightbulbs in a randomly selected batch is shown below. Find the expected value.

A company tracks the number of complaints they receive, where the random variable X is the number of complaints received daily. Find the variance & standard deviation of this distribution.

Identifying Discrete and Continuous Random Variables. In Exercises 5 and 6, refer to the given values, then identify which of the following is most appropriate: discrete random variable, continuous random variable, or not a random variable.

a. IQ scores of statistics students

b. Exact heights of statistics students

c. Shoe sizes (such as 8 or 8½) of statistics students

d. Majors (such as history) of statistics students

e. The number of rolls of a die required for a statistics student to get the number 4

Discrete or Continuous? Is the random variable given in the table from Exercise 1 discrete or continuous? Explain.

Family/Partner Groups of people aged 15–65 are randomly selected and arranged in groups of six. The random variable x is the number in the group who say that their family and/or partner contribute most to their happiness (based on a Coca-Cola survey). The accompanying table lists the values of x along with their corresponding probabilities. Does the table describe a probability distribution? If so, find the mean and standard deviation.

Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

Plane Crashes The table lists causes of fatal plane crashes with their corresponding probabilities.

Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

Online Courses College students are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they take one or more online courses (based on data from Sallie Mae).

Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

Fear of Heights The table lists results from a survey of 285 subjects who were asked, “Are you afraid of heights in tall buildings?” The results are from USA Today.

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

a. Find the probability that in a year, there will be no hurricanes.

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

b. In a 118-year period, how many years are expected to have no hurricanes?

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

a. Find the probability that in a year, there will be 10 hurricanes.

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

b. In a 118-year period, how many years are expected to have 10 hurricanes?

Find the mean of the random variable x described in the preceding exercise.

Is the mean found in the preceding exercise a statistic or a parameter?

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).

a. Find the mean number of deaths per day.