91Ó°ÊÓ

Pyramid Lake is located in Nevada on the Paiute Indian Reservation. This lake is famous for large cutthroat trout. The mean number of trout (large and small) caught from a boat is 0.667 fish per hour (Reference: Creel Chronicle, Vol. \(3,\) No. 2 ). Suppose you rent a boat and go fishing for 8 hours. Let \(r\) be a random variable that represents the number of fish you catch in the 8 -hour period. (a) Explain why a Poisson probability distribution is appropriate for \(r\). What is \(\lambda\) for the 8 -hour fishing trip? Round \(\lambda\) to the nearest tenth so that you can use Table 4 of Appendix II for Poisson probabilities. (b) If you have already caught three trout, what is the probability you will catch a total of seven or more trout? Compute \(P(r \geq 7 | r \geq 3) .\) See Hint below. (c) If you have already caught four trout, what is the probability you will catch a total of fewer than nine trout? Compute \(P(r<9 | r \geq 4) .\) See Hint below. (d) List at least three other areas besides fishing to which you think conditional Poisson probabilities can be applied. Hint for solution: Review item \(6,\) conditional probability, in the summary of basic probability rules at the end of Section \(4.2 .\) Note that $$P(A | B)=\frac{P(A \text { and } B)}{P(B)}$$ and show that in part (b), $$P(r \geq 7 | r \geq 3)=\frac{P[(r \geq 7) \text { and }(r \geq 3)]}{P(r \geq 3)}=\frac{P(r \geq 7)}{P(r \geq 3)}$$

Step by step solution

01

Justify Poisson Distribution

The Poisson probability distribution is suitable for modeling the count of events happening in a fixed interval of time or space when these events occur with a known constant mean rate and independently of each other. Here, the random variable \( r \) describes the number of fish caught, and the events fit the criteria of a Poisson distribution as they describe a number of events (fish caught) happening at a constant rate (mean 0.667 fish/hour) over an 8-hour period.

02

Calculate \( \lambda \) for the 8-hour Period

To find the mean number \( \lambda \) used in the Poisson distribution, multiply the rate of catching fish per hour by the number of hours. From the problem statement, the mean rate is 0.667 fish/hour. For an 8-hour trip: \[ \lambda = 0.667 \times 8 = 5.336. \] Round this to the nearest tenth to simplify computations: \( \lambda \approx 5.3 \).

03

Calculate \( P(r \geq 7 | r \geq 3) \)

We use conditional probability. First, calculate \( P(r \geq 7) \) using the Poisson distribution: this involves summing probabilities for all \( r \geq 7 \). Then, find \( P(r \geq 3) \) by summing probabilities for all \( r \geq 3 \). As per the hint, \( P(r \geq 7 | r \geq 3) = \frac{P(r \geq 7)}{P(r \geq 3)}. \) Utilize Poisson probability tables to obtain these values.

04

Calculate \( P(r

To compute \( P(r < 9 | r \geq 4) \), first determine \( P(r < 9) \) by summing probabilities for \( r \, \) values from 0 to 8. Next, find \( P(r \geq 4) \) by summing probabilities starting from \( r = 4 \). Use the conditional probability formula: \( P(r<9 | r \geq 4) = \frac{P((r<9) \ and \ (r \geq 4))}{P(r \geq 4)} = \frac{P(4 \leq r < 9)}{P(r \geq 4)}. \) The probability tables will give the needed values.

05

Applications of Conditional Poisson Probabilities

Besides fishing, conditional Poisson probabilities can be applied in areas such as traffic flow (e.g., the number of cars passing a checkpoint), call center management (e.g., incoming calls per minute), and email or spam filtering (e.g., receiving emails in a given timeframe). These situations involve counting events occurring at a constant average rate.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring given that another event has already taken place. It's an essential concept in probability theory and particularly useful in situations where additional information is known.

In our fishing scenario, for example, you want to know the probability of catching a total of seven or more fish, given that you've already caught three. This is symbolized as \( P(r \geq 7 | r \geq 3) \). To calculate this, you would use the formula for conditional probability:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

In this instance, the event \( A \) is catching seven or more fish, and \( B \) is having caught at least three fish.

The calculation simplifies to \( \frac{P(r \geq 7)}{P(r \geq 3)} \), using the Poisson probabilities for \( r \geq 7 \) and \( r \geq 3 \). This helps quantify your chances given your prior success.

Mean Rate

The mean rate in a Poisson distribution reflects how frequently an event occurs on average within a given time frame or spatial area. This concept is crucial because it sets the stage for calculating probabilities related to event counts.

In our exercise, the mean rate of catching fish is given as 0.667 fish per hour. To apply this to an 8-hour fishing trip, you multiply 0.667 by 8, yielding a mean of 5.336 fish for the 8-hour period. It is often rounded to a more manageable number for use in Poisson tables, hence, 5.3.

Understanding the mean rate allows you to predict the expected number of events (in this case, fish caught) over your interval of interest. This is pivotal in setting up and interpreting the parameters of your Poisson distribution calculations.

Event Occurrence

In the context of Poisson distribution, event occurrence refers to the actual number of times an event (like catching a fish) happens during a given interval. Each event is assumed to happen independently and with a constant probability over time.

This independence of events is one of the hallmarks of Poisson processes. For instance, the likelihood of catching a fish in one hour may not depend on the catch rate of previous hours, provided the conditions remain the same. This ensures that the distribution can reliably model situations like our fishing trip, where we count fish caught over a span of hours.

With different outcomes ranging from fewer to more catches, Poisson probabilities give a structured way to understand these occurrences using the calculated mean rate.

Count Data

Count data are numerical values that represent the number of occurrences of an event within a fixed criterion of time or space. In our discussion, the random variable \( r \) is the count data representing the number of fish caught.

This kind of data is central to Poisson distribution because it models events counted in discrete units. For example, the count data would be valid if measuring something like the number of emails received in an hour or fish caught in a day.

• This data type is particularly handy in probabilistic modeling because it often follows predictable probability patterns when provided a mean rate.

Exploring count data through Poisson distribution sheds light on expected outcomes and the variability or randomness that could naturally occur, allowing for more informed predictions and decisions in real-world scenarios.