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Understanding expected value is like determining the average outcome if you repeat an experiment over and over. It involves using a probability density function, in this case, a formula representing how CPU time usage is spread over a week.
To find expected value, denoted as \( E(Y) \), we integrate the product of the variable \( Y \) and its PDF over its range. For this problem, the formula becomes:

• \( E(Y) = \int_{0}^{4} y \left( \frac{3}{64} y^2(4-y) \right) \, dy \), which simplifies, upon solving, to finding the integral \( \int_{0}^{4} y^3(4-y) \, dy \).

Integration then requires solving two distinctive parts:

• \( \int_{0}^{4} 4y^3 \, dy \)
• \( \int_{0}^{4} y^4 \, dy \)

Each integral provides part of the whole picture, which when calculated, gives us the expected CPU hours used weekly. This mean reflects not just a guess but a calculable average which is vital in planning resources.
By understanding the expected value, firms can predict their average CPU usage, which can help in budget planning and forecasting potential technological resources needed.

Variance provides a measure of how much the CPU time usage fluctuates around the expected value. In other words, it tells us about the spread or variability of the weekly CPU time. If the variance is large, it means CPU time usage is highly unpredictable.
To calculate the variance of the CPU time, we must first find \( E(Y^2) \), an expectation involving each possible value squared and weighted by probability. The healthy formula is:

• \( E(Y^2) = \int_{0}^{4} y^2 \left( \frac{3}{64} y^2(4-y) \right) \, dy \)

Solving this integral can be intricate, but once we have \( E(Y^2) \), variance \( \text{Var}(Y) \) is determined by subtracting the square of \( E(Y) \) from \( E(Y^2) \).
This difference shows us the extent of variation from the mean, or how spread out the CPU usage is from its expected value. Low variance would suggest that the CPU time stays close to the mean, indicating consistency, whereas high variance suggests more unpredictability.

Integration plays a central role in calculating both expected value and variance in probability density functions. It's a mathematical tool that helps sum up continuous variables over a given interval. In the context of CPU time, integration helps us determine both average values and their fluctuations over time.
As described in the context of solving for expected value and variance, integration takes the form of summing the area under a curve defined by the PDF. It's essential to understand that:

• Integrals such as \( \int_{0}^{4} y^3(4-y) \, dy \) are solved by breaking them into simpler parts for easy calculation.
• In the variance calculation, both \( \int_{0}^{4} 4y^3 \, dy \) and \( \int_{0}^{4} y^4 \, dy \) represent solving individual expressions that contribute to the overall spread of values.

This is part of why integration is powerful: by breaking down complex PDF expressions, it delivers precise insights into the behavior of data over time.
Efficient integration can provide critical reflections of a dataset's central tendencies and dispersions, helping companies make better data-driven decisions.