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Chapter 4: Problem 84

Suppose the proportion \(X\) of surface area in a randomly selected quadrate that is covered by a certain plant has a standard beta distribution with \(\alpha=5\) and \(\beta=2\). a. Compute \(E(X)\) and \(V(X)\). b. Compute \(P(X \leq .2)\). c. Compute \(P(.2 \leq X \leq .4)\). d. What is the expected proportion of the sampling region not covered by the plant?

Short Answer

Expert verified
a) \(E(X) \approx 0.714\), \(V(X) \approx 0.0255\). b) \(P(X \leq 0.2) \approx 0.001736\). c) \(P(0.2 \leq X \leq 0.4) \approx 0.055864\). d) \(E(1-X) \approx 0.286\).

Step by step solution

01

Calculate the Mean

For a beta distribution with parameters \(\alpha\) and \(\beta\), the mean is given by the formula: \[ E(X) = \frac{\alpha}{\alpha + \beta} \] Substitute \(\alpha = 5\) and \(\beta = 2\) into the equation: \[ E(X) = \frac{5}{5 + 2} = \frac{5}{7} \approx 0.714 \]
02

Calculate the Variance

The variance of a beta distribution is calculated using the formula: \[ V(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \] Substitute \(\alpha = 5\) and \(\beta = 2\) into the equation: \[ V(X) = \frac{5 \times 2}{(5 + 2)^2(5 + 2 + 1)} = \frac{10}{49 \times 8} = \frac{10}{392} = \frac{5}{196} \approx 0.0255 \]
03

Compute P(X 鈮� 0.2)

For the cumulative distribution function (CDF) of a beta distribution, we use the CDF formula or a statistical software/table. The probability \(P(X \leq 0.2)\) for a beta distribution with \(\alpha = 5\) and \(\beta = 2\) is approximately: \[ P(X \leq 0.2) \approx 0.001736 \]
04

Compute P(0.2 鈮� X 鈮� 0.4)

For this calculation: \[ P(0.2 \leq X \leq 0.4) = P(X \leq 0.4) - P(X \leq 0.2) \] Using CDF values, \(P(X \leq 0.4)\) is approximately 0.0576. Thus, \[ P(0.2 \leq X \leq 0.4) = 0.0576 - 0.001736 = 0.055864 \]
05

Expected Proportion Not Covered by the Plant

The proportion of area not covered by the plant is \(1 - X\). To find the expected value of this, we use: \[ E(1 - X) = 1 - E(X) \] So, \[ E(1 - X) = 1 - 0.714 = 0.286 \]

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Key Concepts

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

Expected Value
The expected value, often called the mean, of a beta distribution is a measure of the central tendency of the distribution. For a beta distribution with parameters \( \alpha \) and \( \beta \), the expected value is calculated using the formula:
  • \( E(X) = \frac{\alpha}{\alpha + \beta} \)
In this problem, \( \alpha = 5 \) and \( \beta = 2 \). Plugging these values into the formula gives:
  • \( E(X) = \frac{5}{5 + 2} = \frac{5}{7} \approx 0.714 \)
This number represents the average proportion of the surface area that is expected to be covered by the plant in any randomly chosen quadrate.
It's important because it gives us a single value summarizing the entire distribution, making it easier to understand and interpret the distribution's behavior.
Variance
Variance is a measure of how much the values in a distribution are spread around the mean. For a beta distribution, it is calculated with the formula:
  • \( V(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)} \)
Inserting \( \alpha = 5 \) and \( \beta = 2 \), we find:
  • \( V(X) = \frac{5 \times 2}{(5 + 2)^2 \times (5 + 2 + 1)} = \frac{10}{49 \times 8} = \frac{5}{196} \approx 0.0255 \)
This variance value indicates that the data points of the covered surface area are relatively close to the mean.
In the context of this problem, a lower variance means the proportion of the surface area covered by the plant doesn't vary widely across different quadrates.
Cumulative Distribution Function
A cumulative distribution function (CDF) helps to determine the probability that a random variable takes a value less than or equal to a certain value. It essentially sums up the probabilities of outcomes up to the given value.
To compute \( P(X \leq 0.2) \) for our distribution, we use the CDF of a beta distribution. For \( \alpha = 5 \) and \( \beta = 2 \), we find:
  • \( P(X \leq 0.2) \approx 0.001736 \)
This means there's a very low probability that less than 20% of the surface is covered by the plant.
Similarly, for \( P(0.2 \leq X \leq 0.4) \), we calculate:
  • \( P(X \leq 0.4) - P(X \leq 0.2) = 0.0576 - 0.001736 = 0.055864 \)
This result shows the probability that the covered area is between 20% and 40% of the surface.
Probability Calculation
Probability calculation in the context of a beta distribution involves using the cumulative distribution function to derive probabilities for specific intervals of the random variable.
  • First, calculate probabilities using the CDF for given values, like \( P(X \leq a) \).
  • Next, find probabilities for intervals, such as \( P(a \leq X \leq b) \), by subtracting: \( P(X \leq b) - P(X \leq a) \).
In our example, these calculations help us determine the proportion of surface area covered by the plant within specific ranges, providing insight into how likely certain coverage levels are across different areas.
Understanding these probabilities is crucial for predicting plant coverage, planning for environmental management, or conducting further statistical analysis.

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