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The expected value, often symbolized by \( E[X] \), is a fundamental concept in probability and statistics. It represents the average outcome or mean of a random variable over a large number of trials. In the context of a Poisson distribution, the expected value is simply the parameter \( \lambda \), which describes the average number of occurrences in a fixed interval.

In our exercise, the number of diseased trees \( Y \) per acre follows a Poisson distribution with \( \lambda = 10 \). Hence, the expected value \( E[Y] \) is 10. The total cost \( C \), which includes both a fixed charge and a variable charge dependent on the number of diseased trees, can be expressed as \( C = 50 + 3Y \).

Using the property of linearity of expectation, the expected value of the total spraying cost \( C \) can be calculated as follows:

• \( E[C] = E[50 + 3Y] \) which simplifies to \( 50 + 3E[Y] \)
• Substituting \( E[Y] = 10 \), we find \( E[C] = 50 + 3 \times 10 = 80 \)

Thus, the expected total spraying cost per acre is 80 units, which can be interpreted as dollars in this scenario.

The standard deviation is a measure of the amount of variability or spread in a set of values. It is denoted by \( \sigma \) and provides insight into how much the values of a random variable, such as the number of diseased trees, diverge from the mean.

For a Poisson-distributed variable like \( Y \), the variance \( \text{Var}(Y) \) is equal to its mean \( \lambda \). In this case, \( \lambda = 10 \), so the variance of \( Y \) is 10. The standard deviation \( \sigma_Y \) is then the square root of the variance: \( \sigma_Y = \sqrt{10} \).

To find the standard deviation of the total cost \( C = 50 + 3Y \), apply the rule that the standard deviation of a constant multiplied by a variable is the constant times the standard deviation of the variable:

• \( \sigma_C = 3 \times \sigma_Y = 3 \times \sqrt{10} \)
• This results in \( \sigma_C \approx 9.49 \)

The standard deviation of \( C \) tells us how the total costs likely deviate from the mean cost of 80 dollars.

The probability interval refers to the range in which a random variable falls with a specified probability. For symmetric distributions, intervals around the mean can provide insight into typical values observed.

In the exercise, we assess the interval within which we expect the total cost \( C \) to fall with at least 75% probability. Assuming a normal approximation of the Poisson distribution for simplicity, an interval of about mean \( \pm \sqrt{2}(\sigma_C) \) often covers approximately 75% of the distribution.

Given:

• Mean \( E[C] = 80 \)
• Standard deviation \( \sigma_C \approx 9.49 \)

We compute the interval:

• \( 80 \pm \sqrt{2} \times 9.49 \)
• This approximation gives \( 80 \pm 13.43 \)
• Resulting in the interval \([66.57, 93.43]\)

Thus, the costs are expected to lie within this range for about 75% of instances, considering the Poisson nature of the data and the normal approximation applied here.