Problem 15 - Applied Calculus

Question

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. All yields from 0 to 100 kg are equally likely; the field never yields more than \(100 \mathrm{kg}\).

Step-by-Step Solution

Answer

The density function is constant: \( f(x) = \frac{1}{100} \) for \( 0 \leq x \leq 100 \).

Step 1: Understand the Problem

We need to graph a density function that represents the crop yield, which is uniformly distributed between 0 and 100 kg.

Step 2: Identify the Type of Distribution

The problem describes a situation where all yields from 0 to 100 kg are equally likely. This indicates a uniform distribution.

Step 3: Define the Uniform Distribution

For a uniform distribution over an interval [a, b], the probability density function (pdf) is defined as \( f(x) = \frac{1}{b-a} \) for \( a \leq x \leq b \), and \( f(x) = 0 \) otherwise.

Step 4: Calculate the Probability Density Function

In this case, the interval is [0, 100]. Thus, the pdf is given by \( f(x) = \frac{1}{100-0} = \frac{1}{100} \) for \( 0 \leq x \leq 100 \).

Step 5: Create the Graph of the Density Function

Draw a horizontal line at \( y = \frac{1}{100} \) from \( x = 0 \) to \( x = 100 \). This line represents the pdf, where the height is constant across the interval.

Key Concepts

Probability Density Function

The concept of a Probability Density Function (pdf) is central to the topic of probability and statistics. In essence, a pdf helps us understand the likelihood of a particular outcome happening within a given interval. Unlike discrete probabilities that deal with separate events, pdfs are used for continuous data.

For continuous uniform distributions, such as the one described in the crop yield example, the pdf is a simple constant value across the interval of interest, meaning all outcomes are equally likely. The probability density function, therefore, takes the form of a horizontal line over this interval.

The uniform distribution for crop yield from 0 to 100 kilograms means every yield value is equally probable, resulting in a pdf defined as:

• For any value, say \( x \), within the interval \( [0, 100] \), the pdf \( f(x) \) is \( \frac{1}{100} \).
• Beyond this interval, the pdf is zero, indicating no probability of occurring.

Crop Yield

Crop yield in statistical terms is typically modeled according to the nature of the distribution the yield follows. In practice, this involves variables like soil quality, weather, and seed type.

In this exercise, yield is modeled as uniformly distributed between 0 and 100 kilograms. This means each possible yield within this range is equally likely. Such modeling assumes that there is no systematic variance between different possible yields in this scenario.

• The assumption of a maximum yield with a given upper bound (100 kg here) is quite common to simplify calculations and is helpful in agricultural planning.
• Real-world yield data often require more complex models because of various influencing factors, but uniform distribution provides a baseline understanding.

Graphing Functions

Graphing functions is an essential skill in visualizing mathematical relations and distributions. In the context of a probability density function, graphing involves representing how probability is spread out over an interval.

For the uniform distribution example of the crop yield:

• The graph involves a straight line parallel to the x-axis, indicating a constant probability across the interval.
• The line begins at \( x = 0 \) and ends at \( x = 100 \), remaining at a height of \( y = \frac{1}{100} \).

This visual representation helps intuitively grasp that all possible outputs within the interval are equally likely. When graphing, ensure that axes are correctly labeled - the x-axis for potential yields and the y-axis for probability densities.