Question

In this lab, you have investigated six of the most important distributions in probability theory: You should now have a good idea of when to expect these distributions to appear: For the random variables below, indicate whether you would expect the distribution to be best described as geometric; binomial, Poisson; exponential, uniform, or normal: We do not have data; so you will not to use the computer for these questions For each item, give a brief explanation of your answer: A one-sentence explanation should be sufficient: The number of goals that a team scores in a hockey game: The time of day that the next major earthquake occurs in Southern California. The number of minutes before a store manager gets her next phone call The number of 3's that appear in 20 rolls of a die The number of days out of the next 10 that a stock will go up_ The amount of time before the next customer arrives in a store. The number of particles that a radioactive substance emits in the next two seconds. The number of free throws that a basketball player needs to make before missing one

Answer 1

In these scenarios, the Poisson distribution is used for count-based events within fixed intervals, the exponential distribution is best for timing until an event occurs, the binomial distribution is appropriate for fixed trials with two outcomes, and the geometric distribution applies to trials until a single failure.

Step-by-step explanation:

For the random variables presented, the corresponding distributions can be identified as follows:

• The number of goals that a team scores in a hockey game is likely a Poisson distribution because goals happen independently over the time of the game with a known average rate.
• The time of day that the next major earthquake occurs in Southern California can be modeled by an exponential distribution since it deals with the amount of time until an event occurs.
• The number of minutes before a store manager gets her next phone call is also an example of an exponential distribution, representing the time until the next event.
• The number of 3's that appear in 20 rolls of a die could be modeled by a binomial distribution, because there are a fixed number of independent trials with two possible outcomes.
• The number of days out of the next 10 that a stock will go up could also follow a binomial distribution, assuming each day is independent and has the same probability of the stock going up.
• The amount of time before the next customer arrives in a store is an example of an exponential distribution, with a focus on the time until the next event occurs.
• The number of particles that a radioactive substance emits in the next two seconds would be modeled by a Poisson distribution, considering a known average rate of emission over a time interval.
• The number of free throws that a basketball player needs to make before missing one is a geometric distribution, as it measures the number of trials until the first failure.