4.6 Poisson Distribution

A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let X = the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is s…

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

a. What is the interval of interest?

b. What is the average number of times the news reporter says "uh" during one broadcast?

c. Let X = ____________. What values does X take on?

d. The probability question is P(______).

According to Baydin, an email management company, an email user gets, on average, 147 emails per day. Let X = the number of emails an email user receives per day. The discrete random variable X takes on the values x = 0, 1, 2 …. The random variable X has a Poisson distribution: X ~ P(147). The mean is 147 emails.

1. What is the probability that an email user receives exactly 160 emails per day?
2. What…

Text message users receive or send an average of 41.5 text messages per day.

1. How many text messages does a text message user receive or send per hour?
2. What is the probability that a text message user receives or sends two messages per hour?
3. What is the probability that a text message user receives or sends more than two messages per hour?

On a certain day in May starting at 4:30 PM, the probability of low seismic activity for the next 48 hours in Alaska was reported as about 1.02%. Use this information for the next 200 days to find the probability that there will be low seismic activity in ten of the next 200 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close?

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

What values does X take on?

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

What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day.

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

What is the probability that the store will have fewer than 12 customers in the first two hours?

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

Assume the event occurs independently in any given day. In words, define the random variable X.

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

What values does X take on?

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

Is it likely that there will be no teens killed from motor vehicle injuries on any given day in the U.S? Justify your answer numerically.

The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let X = the number of calls received at noon.

1. Find the mean and standard deviation of X.
2. What is the probability that the office receives at most six calls at noon on Monday?
3. Find the probability tha…

A manufacturer of Christmas tree light bulbs knows that 3% of its bulbs are defective. Find the probability that a string of 100 lights contains at most four defective bulbs using both the binomial and Poisson distributions.

The average number of children a Spanish woman has in her lifetime is 1.47. Suppose that one Spanish woman is randomly chosen.

1. In words, define the Random Variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(_____,_____)
4. Find the probability that she has no children.
5. Find the probability that she has fewer children than the Spanish average.
6. Find the probability th…

The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem, but only use one distribution to solve the problem.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(_____,_…

The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. Suppose that 100 people with tax returns over $25,000 are randomly picked. We are interested in the number of people audited in one year. Use a Poisson distribution to anwer the following questions.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X…

On average, Pierre, an amateur chef, drops three pieces of egg shell into every two cake batters he makes. Suppose that you buy one of his cakes.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(_____,_____)
4. On average, how many pieces of egg shell do you expect to be in the cake?
5. What is the probability that there will not be any p…

Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is ten. We are interested in the number of times her cats wake her up each week.

Find the probability that her cats will wake her up no more than five times next week.

1. 0.5000
2. 0.9329
3. 0.0378
4. 0.0671