Poisson distribution: notes and practice questions

A factory produces specialized electronic components. The total "quality score" of a component, $T$, is a combination of scores from three independent inspection stages:

• Stage 1: Automated visual inspection. The score $X$ from this stage has an expectation of $2.5$ and a standard deviation of $0.7$.

• Stage 2: Manual functional test. A batch of $8$ critical functions are tested, and the score $Y$ is the number of functions that pass. Each function has a $0.35$ probability of passing, independently.

• Stage 3: Environmental stress test. The score $Z$ from this stage, representing the number of successful stress cycles, follows a Poisson distribution with a mean of $4.2$.

The overall quality score for a component is given by $T = X + 2Y + Z$.

Calculate the expected value and variance of the total quality score $T$.

Given that the distribution of $T$ can be approximated by a Normal distribution, find the probability that a randomly selected component has a total quality score between $10$ and $14$ (inclusive of $10$, exclusive of $14$).

The factory manager wants to ensure that the mean total quality score of a sample of $n$ components is within $0.8$ units of the true mean, with a probability of at least $0.95$. Find the minimum sample size $n$ required.

A bank manager observes that the number of customers arriving at their main ATM during peak hours follows a Poisson distribution with a mean of 18 customers per hour.

To encourage more foot traffic, the bank launches a new promotional campaign. The manager wants to test if this campaign has increased the number of ATM arrivals. They decide to monitor the ATM for a single 4-hour peak period and use a 5% level of significance for their test.

(a) State the null and alternative hypotheses for the test.

(b) Find the probability that the bank manager will make a Type I error in their test conclusion.

During the 4-hour observation period, the bank manager recorded 85 customer arrivals at the ATM.

(c) State the bank manager's conclusion to the test. Justify your answer.

A call centre receives calls at an average rate of $1.8$ calls per minute during a specific period of the day.

The number of calls received in a given minute can be modelled by a Poisson distribution.

(a) Find the probability that the call centre receives exactly one call in a randomly selected minute.

(b) Find the probability that the call centre receives at least one call in a randomly selected minute.

The call centre supervisor observes the calls received over a period of $4$ independent minutes.

(c) Find the probability that a total of six calls are received during these $4$ minutes.

(d) Find the probability that exactly one call is received in each of these $4$ minutes.

(e) Find the probability that at least one call is received in exactly $2$ of the $4$ minutes.

A high-end watch manufacturer implements a quality control system. The average number of defective watches, $D$, in a standard production batch is related to the quality control score, $Q$, by the equation $\log_{10}D = k - Q$, where $k \in \mathbb{R}$.

For a quality control score of $Q = 2.5$, the manufacturer observes an average of $200$ defective watches per standard batch.

(a) Find the value of $k$.

(b) The equation relating $D$ and $Q$ can also be expressed in the form $D = \frac{m}{10^Q}$.

Find the value of $m$.

(c) The manufacturer aims for an extremely high quality control score of $Q = 7.8$.

Find the average number of defective watches in a standard batch for this quality score.

(d) The number of defective watches in a production run can be modelled by a Poisson distribution. The average number of defective watches in a standard batch (from part (c) ) represents the mean for a single production unit.

A large order consists of $250$ such standard production batches. Find the probability that this entire large order has no defective watches.

(a) The number of customer service emails received by 'TechSolutions' per hour follows a Poisson distribution with a mean of $2.8$. Using this model, find the probability that TechSolutions receives exactly $3$ emails in a given hour.

(b) Over a period of $4$ consecutive hours, find the probability that TechSolutions receives:

(i) exactly $10$ emails.

(ii) emails during the first and third hour only (i.e., at least one email in the first hour, no emails in the second hour, and at least one email in the third hour).

(c) Over a working day of $10$ hours, find the probability that there are exactly $2$ hours during which TechSolutions receives no emails.

(d) TechSolutions expands its operations and opens a new department, 'SupportPlus', which also receives emails. The number of emails received by each 'SupportPlus' agent per hour follows a Poisson distribution with a mean of $1.5$. Assuming these are independent of the main TechSolutions department and each other, determine the least number of SupportPlus agents required so that the total probability of receiving at least $25$ emails across all departments (main TechSolutions and the SupportPlus agents) in an hour is greater than $0.15$.

(a) The manager of "The Daily Grind" coffee shop suggested that the number of customers arriving at the shop during a 5-minute interval can be modelled by a Poisson distribution.

Suggest two observations that the manager may have made that led him to suggest this model.

(b) Now assume that the model is valid and that the mean number of customers arriving at the shop during a 5-minute interval is $1.5$.

The manager observes customer arrivals during a 15-minute interval.

Calculate the probability that exactly 6 customers arrive during this 15-minute interval.

(c) Using the same model as in part (b), find the probability that fewer than 4 customers arrive during a 15-minute interval.

(d) Find the probability that in four consecutive 5-minute intervals, at least one customer arrives in each interval.

(e) Following a new marketing campaign, the manager wished to determine whether the mean number of customers arriving during a 5-minute interval had increased.

State the hypotheses for the test.

(f) Find the critical region for the test at the 5% significance level.

(g) Given that the mean number of customers per 5-minute interval has actually risen to $2.5$, find the probability that the manager makes a Type II error.

The number of customer complaints received by an online retailer per day is modelled by a Poisson distribution with a mean of $0.85$ complaints.

Under this model, calculate the probability that there are at least $2$ complaints in a particular day.

Calculate the probability that there will be exactly $4$ complaints in a particular $3$-day period.

The retailer wants to determine the probability that, in a $7$-day week, fewer than $2$ complaints occur in a day on exactly $3$ occasions. It assumes that the daily occurrence of complaints is independent of the day on which these occur.

State the appropriate model that the retailer should use to determine this probability.

A customer service department receives contacts from two independent sources: phone calls and emails. The number of phone calls received per hour can be modelled by a Poisson distribution with a mean of $5.5$. The number of emails received per hour can be modelled by a Poisson distribution with a mean of $3.8$.

(a)(i) Find the probability that the department receives at most $20$ phone calls in a $3$-hour period.

(a)(ii) Find the probability that the department receives a total of more than $12$ contacts (phone calls and emails combined) in one hour.

Each phone call takes an average of $4$ minutes to handle, and each email takes an average of $7$ minutes to handle. Let $M$ be the total time (in minutes) spent handling contacts in one hour.

(b)(i) Find $E(M)$.

(b)(ii) Find $Var(M)$.

(c) State one reason why the distribution of $M$ cannot be a Poisson distribution.

(d) The customer service department operates for an $8$-hour shift. Use the Central Limit Theorem to find the probability that the mean time spent per hour handling contacts during this shift is greater than $45$ minutes.

A small artisanal bakery, "The Sweet Spot", specializes in custom-designed celebration cakes. On average, they receive $28$ custom cake orders per week. The number of orders received can be modelled by a Poisson distribution.

The head baker can only complete a maximum of $16$ custom cakes in a 3-day period due to the intricate nature of the designs.

(a) Calculate the probability that some custom cake orders are not fulfilled in a 3-day period.

A supervisor is deciding whether to employ an additional baker for "The Sweet Spot". They monitor the custom cake orders for a random 3-day period each month for $12$ months.

(b) Find the probability that all custom cake orders are fulfilled on at least $10$ of these $12$ months.

(a) Mr. Lee, the owner of "Sweet Delights" bakery, recorded the number of Mooncakes sold each day for a sample of $30$ days. The results are shown in the table below.

(a.i) Find the mean and variance for this sample data.

(a.ii) Hence, state why Mr. Lee might believe that the daily sales of Mooncakes follow a Poisson distribution.

(b) State one assumption that Mr. Lee needs to make about the sales of Mooncakes to support his belief that it follows a Poisson distribution.

(c) Mr. Lee knows from his historic sales records that the bakery sells an average of $3.9$ Mooncakes each day. The following table shows the expected frequency of Mooncakes sold each day during a $100$-day period, assuming a Poisson distribution with mean $3.9$.

Find the value of a, of b, and of c. Give your answers to 3 decimal places.

(d) Mr. Lee decides to carry out a $\chi^2$ goodness of fit test at the $5\%$ significance level to see whether the daily sales of Mooncakes follow a Poisson distribution with mean $3.9$. He collects observed frequencies for $100$ days, which are given in the table below.

(d.i) Write down the number of degrees of freedom for his test.

(d.ii) Perform the $\chi^2$ goodness of fit test and state, with reason, a conclusion.

(e) Mr. Lee claims that a new social media advertising campaign, costing $250$ THB per day, will increase the number of Mooncakes sold. However, his business partner, Ms. Chen, claims that the advertising will not increase the bakery's overall profit.

Ms. Chen agrees to run the campaign for the next $40$ days. During that time, Mr. Lee records that the bakery sells a total of $180$ Mooncakes, with a profit of $45$ THB on each Mooncake sold.

Mr. Lee wants to carry out an appropriate hypothesis test to determine whether the number of Mooncakes sold during the $40$ days increased when compared with the historic sales records (mean $3.9$ Mooncakes per day).

By finding a critical value, perform this test at a $5\%$ significance level.

(f) Hence state the probability of a Type I error for this test.

(g) By considering the claims of both Mr. Lee and Ms. Chen, explain whether the advertising campaign was beneficial to the bakery.

The number of customers arriving at a popular bakery during peak hours can be modelled by a Poisson distribution. During a 30-minute busy period, customers arrive at a mean rate of 7.8 customers.

(a)
(i) Find the probability that exactly 6 customers arrive during a 30-minute busy period.

(a)
(ii) Find the most likely number of customers that would arrive during a 30-minute busy period.

(a)
(iii) Given that more than 8 customers arrive during a 30-minute busy period, find the probability that exactly 10 customers arrive.

(b) During quiet periods of the day, customers arrive at a mean rate of 2.1 customers every 15 minutes.

Find the probability that during a period of 45 minutes, of which the first 30 minutes is busy and the next 15 minutes is quiet, exactly 5 customers arrive.