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An unbiased estimator is a statistical term used to describe an estimate of a parameter, such as the mean, that is precisely equal to the expected value of the parameter being estimated. In simpler terms, it means that over numerous samples, the average of the estimates will converge to the true value of the parameter.

In the context of the Poisson distribution, where we are analyzing the number of minicomputer breakdowns per week, the unbiased estimator for the parameter \( \lambda \) is the mean of the sample, denoted as \( \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i \). This is because the expected value of \( \bar{Y} \) is \( \lambda \), satisfying the condition of being an unbiased estimator.

Using \( \bar{Y} \) achieves an accurate representation of \( \lambda \) without systematic error, which is the key characteristic of unbiased estimators and essential for making reliable inferences in statistical analysis.

The expected value is a fundamental concept in probability that represents the average outcome of a random variable if an experiment is repeated numerous times. For the Poisson distribution, this involves determining the expected number of breakdowns, as well as any associated costs.

In this exercise, the cost function is given by \( C = 3Y + Y^2 \). To calculate the expected cost \( E(C) \), we break it down into two components: the expected value of \( Y \), which is \( \lambda \) for a Poisson distribution, and the expected value of \( Y^2 \), which is \( \lambda + \lambda^2 \).

Thus, the formula for the expected cost becomes:

• \( E(C) = 3E(Y) + E(Y^2) = 3\lambda + (\lambda + \lambda^2) \)
• Simplifying gives \( E(C) = 4\lambda + \lambda^2 \)

Understanding expected value is crucial as it provides insights into the average long-term cost of breakdowns, enabling effective budgeting and policy-making.

A cost function relates a certain cost to a variable, in this case, the number of breakdowns \( Y \). Given the Poisson distribution's nature and the mean \( \lambda \), the cost function is expressed as \( C = 3Y + Y^2 \), which accounts for both fixed and quadratic components of cost.

To find an unbiased estimator for the expected cost, similar reasoning for \( \lambda \) is utilized. Knowing that \( \bar{Y} \) is an unbiased estimator for \( \lambda \), we substitute to estimate \( 4\lambda + \lambda^2 \) using our sample's average criteria.

The resulting function, \( 4\bar{Y} + (\bar{Y})^2 \), serves as an unbiased estimator of \( E(C) \):

• \( 4\bar{Y} \) estimates \( 4\lambda \)
• \( (\bar{Y})^2 \) estimates \( \lambda^2 \)

This approach helps in assessing anticipated repair costs using available sample data, guiding future forecasts and decisions related to maintenance expenses.