Problem 15 - Applied Calculus

Question

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. A drought makes low yields most common, and there is no yield greater than \(30 \mathrm{~kg}\).

Step-by-Step Solution

Answer

The density function can be represented by a horizontal line at height \(\frac{1}{30}\) from 0 to 30 on the x-axis.

Step 1: Understand the Problem Context

We are asked to graph a density function for crop yield under drought conditions. The crop yields are most commonly low and do not exceed 30 kg. We need a function representing this condition.

Step 2: Choose a Suitable Probability Distribution

Given the problem context, we need a continuous probability distribution where low yields are most frequent, suggesting a left-skewed distribution. A simple model for this is the Uniform Distribution on the interval \([0, 30]\).While a more precise model would be slightly skewed, this is often used for simplicity given no further information.

Step 3: Define the Density Function

For a Uniform Distribution over an interval \([a, b]\), the probability density function (pdf) is given by: \[f(x) = \frac{1}{b-a}, \text{ for } a \leq x \leq b\] For our crop yield from \([0, 30]\), this simplifies to: \[f(x) = \frac{1}{30}, \text{ for } 0 \leq x \leq 30\].

Step 4: Create the Graph

The graph of the density function is a horizontal line at \(f(x) = \frac{1}{30}\) starting at 0 and ending at 30 on the x-axis, representing the range of possible yields. The graph remains at zero outside this interval.

Key Concepts

Crop Yield

Crop yield refers to the amount of crop (in kilograms) that is harvested per unit area. Under normal conditions, various factors such as soil fertility, weather, and agricultural practices influence the yield. However, in this exercise, the yield is affected by a drought. This drought scenario implies that most of the fields will have low productivity.

When analyzing crop yield, identifying and understanding the underlying conditions is crucial. For instance, in cases of drought, it becomes challenging for crops to obtain the necessary nutrients and water required for growth. As a result, plants produce lower yields.

This serves as a real-life illustration of how external factors can significantly affect agricultural outputs. Therefore, when predicting and planning for crop yield, it's essential to consider environmental factors and their consequences.

Uniform Distribution

The uniform distribution is a type of probability distribution where each outcome in a certain range is equally likely to occur. This leads to a constant probability density function (pdf). In this exercise, it is used to model the crop yield under drought conditions from 0 to 30 kilograms.

In a continuous uniform distribution over an interval \(a, b\), the pdf is defined as:

• \(f(x) = \frac{1}{b-a}\)
• For the range \(a \leq x \leq b\)

For example, given that crop yields cannot exceed 30 kg, the uniform distribution over the interval \(0, 30\) means that every yield within this range is equally probable. The pdf for this scenario is a horizontal line, because every outcome has the same chance of occurring.

This approach is a simplified way to understand distributions in situations where each output is equally likely, especially when more detailed information is unavailable.

Probability Distribution

A probability distribution describes how the values of a random variable are distributed. It helps to understand and predict the likelihood of different outcomes. There are several types of probability distributions, but in this exercise, the focus is on the continuous probability distribution.

The continuous probability distribution applies to scenarios where the possible outcomes fall within a range that includes potentially infinite values. It is characterized by a probability density function (pdf), which defines the likelihood of the random variable falling within a particular interval.

• The integral of the pdf over the entire space equals 1.
• For a particular value, the pdf can be viewed as a "height" that indicates the density of probability at that point.

In the context of crop yields during a drought, the uniform distribution is employed to represent the probability across the interval from 0 to 30 kg. Even though actual yields are likely to be low, without additional specifics, the uniform model offers a straightforward, understandable representation.

Understanding probability distributions like this one is central to statistics, as they help to anticipate probabilities in real-world scenarios.