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At Burnt Mesa Pueblo, in one of the archaeological excavation sites, the artifact density (number of prehistoric artifacts per 10 liters of sediment was 1.5 (Source: Bandelier Archaeological Excavation Project: Summer 1990 Excavations at Burnt Mesa Pueblo and Casa del Rito, edited by Kohler, Washington State University Department of Anthropology). Suppose you are going to dig up and examine 50 liters of sediment at this site. Let \(r=0,1,2,3, \ldots\) be a random variable that represents the number of prehistoric artifacts found in your 50 liters of sediment. (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) Compute the probabilities that in your 50 liters of sediment you will find two prehistoric artifacts, three prehistoric artifacts, and four prehistoric artifacts. (c) Find the probability that you will find three or more prehistoric artifacts in the 50 liters of sediment. (d) Find the probability that you will find fewer than three prehistoric artifacts in the 50 liters of sediment.

Step by step solution

01

Explanation of Poisson Distribution Applicability

The Poisson distribution is suitable for modeling the number of events occurring within a fixed interval of time or space, where these events happen with a known constant mean rate and independently of previous events. In this case, the events are the discovery of prehistoric artifacts in 50 liters of sediment. The mean artifact density is given as 1.5 per 10 liters, so for 50 liters, the expected number 21;1;1;2;1;, and thus Poisson distribution is appropriate.

02

Determining the Parameter and Writing the Probability Formula

The mean number of artifacts in 50 liters is 4=1.5 10/liters * 50/liters 2=1152=7.5. This mean is 11;1;; thus . The probability distribution function for a Poisson random variable 2X2 with parameter c5=7.51; is 1;,1;=1;2,1;1-ce;//d-sub25times;2;

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Key Concepts

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

Probability Distribution

A probability distribution provides a blueprint to describe the likelihood of different outcomes in an experiment. It is a core concept in statistics where each outcome is assigned a probability—a number between 0 and 1—indicating its chance of occurring. When dealing with discrete events that happen independently over a fixed space or interval, such as the number of found artifacts in a sediment, the Poisson distribution can be very useful.
In essence, the Poisson distribution helps us model scenarios where events occur continuously and independently. In the artifact example, we are interested in knowing the possible number of artifacts discovered during excavations, and a probability distribution describes this variable's behavior efficiently.

Random Variable

A random variable is a variable whose values depend on the outcomes of a random phenomenon. It acts as a numerical representation of the outcomes of a probability experiment. In our exercise, the random variable, denoted as \( r \), represents the number of artifacts found in 50 liters of sediment. Each potential number of artifacts is an outcome resulting from digging.

• Random variables can be discrete, like in this exercise, where the values are countable (0, 1, 2, ... artifacts found).
• The random variable \( X \) is typically defined with a specific probability distribution, in this case, the Poisson distribution.

This approach allows us to quantify and express uncertainty systematically, making predictions and probability calculations possible.

Mean Rate

The mean rate, often denoted by \( \lambda \), is crucial in defining a Poisson distribution. It represents the average number of occurrences (events) in a fixed interval. For our archaeological site, this is the average number of artifacts expected to be found in a given volume of sediment.

• We start with an artifact density of 1.5 per 10 liters.
• Therefore, in 50 liters, the mean rate \( \lambda \) is calculated by multiplying the density 1.5 by 5 (since 50 liters is five times 10 liters), giving \( \lambda = 7.5 \).

This parameter \( \lambda \) lays the groundwork for determining the probability of different numbers of artifacts being found, according to the Poisson model.

Probability Calculation

Once we have \( \lambda \), we can calculate probabilities for various numbers of artifacts using the Poisson probability function. The formula is:
\[ P(X = k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!} \]
Here, \( k \) is the number of occurrences we are interested in (like finding two artifacts). The only variable component in our scenario is \( k \), while \( \lambda = 7.5 \) and \( e \) is the base of the natural logarithm.

• For example, to find the probability of uncovering exactly two artifacts, substitute \( k = 2 \) into the formula.
• Similarly, to calculate the likelihood of finding three or four artifacts, respectively, we use \( k = 3 \) and \( k = 4 \).

These calculations allow archaeologists and statisticians to make informed predictions and decisions about artifact discoveries in other areas of the excavation sites.