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Ecology: River Otters In his doctoral thesis, L. A. Beckel (University of Minnesota, 1982 ) studied the social behavior of river otters during the mating season. An important role in the bonding process of river otters is very short periods of social grooming. After extensive observations, Dr. Beckel found that one group of river otters under study had a frequency of initiating grooming of approximately \(1.7\) for every 10 minutes. Suppose that you are observing river otters for 30 minutes. Let \(r=0,1,2, \ldots\) be a random variable that represents the number of times (in a 30 -minute interval) one otter initiates social grooming of another. (a) Explain why the Poisson distribution would be a good choice for the probability distribution of \(r\). What is \(\lambda\) ? Write out the formula for the probability distribution of the random variable \(r .\) (b) Find the probabilities that in your 30 minutes of observation, one otter will initiate social grooming four times, five times, and six times. (c) Find the probability that one otter will initiate social grooming four or more times during the 30 -minute observation period. (d) Find the probability that one otter will initiate social grooming fewer than four times during the 30 -minute observation period.

Step by step solution

01

Reasoning for Poisson Distribution

The Poisson distribution is used for counting the number of times an event happens in a fixed interval of time or space, when these events happen with a known average rate, and independently of the time since the last event. The given scenario is about counting the number of grooming instances in 30 minutes, which fits this description well, as grooming events happen with a known average frequency and are independent.

02

Determining λ

The average frequency given is 1.7 grooming instances per 10 minutes. For 30 minutes, the average rate, λ, would be 3 times that, since 30 minutes is three 10-minute intervals. Thus, λ = 1.7 × 3 = 5.1.

03

Poisson Distribution Formula

The probability mass function of the Poisson distribution is given by the formula: \[ P(r; λ) = \frac{λ^r e^{-λ}}{r!} \] where λ = 5.1, r is the number of occurrences, and e is the base of the natural logarithm.

04

Finding Probability for 4 Grooming Instances

To find the probability that one otter initiates social grooming 4 times in 30 minutes, substitute r = 4 into the Poisson formula:\[ P(4; 5.1) = \frac{5.1^4 e^{-5.1}}{4!} \].Calculate the result:

05

Finding Probability for 5 Grooming Instances

For r = 5, plug into the formula:\[ P(5; 5.1) = \frac{5.1^5 e^{-5.1}}{5!} \].Calculate to get the probability.

06

Finding Probability for 6 Grooming Instances

For r = 6, use the formula:\[ P(6; 5.1) = \frac{5.1^6 e^{-5.1}}{6!} \]. Compute this probability.

07

Probability of 4 or More Grooming Instances

Calculate the probability of 4 or more occurrences by summing the probabilities for r = 4, 5, 6,... to infinity:\[ P(r \geq 4) = P(4) + P(5) + P(6) + dots \].Since Poisson probabilities diminish quickly, only a few terms are likely needed in practice.

08

Probability of Fewer Than 4 Grooming Instances

Find the probability of fewer than 4 instances by summing the probabilities for r = 0 to 3:\[ P(r < 4) = P(0) + P(1) + P(2) + P(3) \].

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Key Concepts

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

Random Variable

A random variable is a central concept in probability theory. It represents a numerical outcome of a random phenomenon. In the context of the river otters scenario, the random variable, denoted by \( r \), is used to count the number of times social grooming is initiated by an otter within a given period, like 30 minutes.
This helps us translate the outcome of a random event (e.g., grooming occasions) into numerical data for further analysis. Random variables can either be discrete (taking specific values) or continuous (taking any value within a range), and in scenarios like this, we often deal with discrete random variables as the number of occurrences is countable.

• Example: If grooming occurs 2 times in 30 minutes, then \( r = 2 \).
• It simplifies the process of linking probabilities to real-world events.

Probability Mass Function

A Probability Mass Function (PMF) is a function that gives us the probability of each possible value of a discrete random variable. Each outcome has a probability associated with it, and these probabilities are derived using the PMF. In Poisson distribution, the PMF is used to find the likelihood of each number of occurrences over a fixed interval.
The PMF for the Poisson distribution is expressed as:\[P(r; \lambda) = \frac{\lambda^r e^{-\lambda}}{r!}\]where \( \lambda \) is the average rate of occurrence and \( e \) is the mathematical constant approximated as 2.71828.

• The PMF allows us to calculate probabilities for different numbers of events, such as exactly 4 grooming instances in 30 minutes.
• It provides a structured way to handle and analyze discrete data.

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In the case of river otters, the Poisson distribution is chosen because it suits circumstances where events occur independently and with a known average rate.
With a Poisson distribution, once the average rate (\( \lambda \)) is established, we can determine the probability of any given number of events happening in a fixed time-lapse by using the PMF. In this specific case, \( \lambda = 5.1 \) as we have calculated from 1.7 grooming sessions per 10 minutes multiplied by 3 (for 30 minutes total).

• This distribution helps model many real-world scenarios involving rare events.
• It is particularly useful for events that may happen with low probability in a larger population size or over a longer time interval.

Statistical Analysis

Statistical analysis involves collecting, reviewing, and drawing conclusions from data. When working with probability distributions such as the Poisson distribution, statistical analysis sets the framework for understanding how data behaves and predicting future outcomes. By calculating the probabilities of different numbers of grooming events, we gain insight into how often such events happen.
To perform statistical analysis:

• Determine the central measure (\( \lambda \)) and use it with a PMF to calculate specific event probabilities.
• Sum probabilities to find cumulative values, like the chance of encountering fewer than or more than a set number of events.
• Apply these probabilities to make predictions or conduct tests about behavior patterns.

Statistical analysis helps bring order and comprehension to what might at first appear as random data.