5. Binomial Distribution & Discrete Random Variables

A student is analyzing different types of variables in a statistics class. Which of the following below is a discrete random variable?

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.

Given that $x$ has a Poisson distribution with parameter $\lambda$, which of the following is the correct expression for the probability that $x$ equals $k$?