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The expected value, often referred to as the mean, is a fundamental concept in probability and statistics. It provides a measure of the central tendency of a random variable. In simpler terms, it's what we "expect" to happen on average over multiple trials. When dealing with a uniformly distributed random variable, calculating the expected value is straightforward.

For a uniform distribution over an interval \[a, b\], every outcome within the interval has an equal probability of occurring, and the expected value is the midpoint of the interval. Mathematically, this can be expressed as:

• \( E[Z] = \frac{a + b}{2} \)

In our context, both \({X}\) and \({Y}\) are uniformly distributed over the interval \[L-A, L+A\]. Thus, the expected value for both variables \((E[X]\) and \({E[Y]})\) is \({L}\), because:

• \( E[X] = E[Y] = \frac{(L-A) + (L+A)}{2} = L \)

These expected values play a crucial role in determining the expected area of the rectangle, as they help us understand the average length expected for each side.

The area of a rectangle is calculated by multiplying its length by its width. In mathematical terms, if one side (length) is represented by \({X}\) and the other (width) by \({Y}\), the area \({A}\) is computed as:

• \( A = X \times Y \)

When these side lengths are random variables, the concept of expected area becomes important.

The expected area is given by the product of the expected values of the lengths of the sides. For independent variables, the rule simplifies to multiplying their individual expected values:

• \( E[XY] = E[X] \times E[Y] \)

Since we know that \({E[X] = L}\) and \({E[Y] = L}\), the expected area is calculated as:

• \( E[XY] = L \times L = L^2 \)

This represents the average area you can expect, even when the side lengths vary due to measurement errors.

In probability and statistics, independence between random variables is a key concept. Two variables are said to be independent if the occurrence of one does not affect the probability of occurrence of the other. This concept is crucial when calculating combined probabilities or expected values, as it simplifies computations significantly.

When dealing with independent random variables \({X}\) and \({Y}\), especially in the context of expected values, the following property holds true:

• \( E[XY] = E[X] \times E[Y] \)

This property is a direct result of independence and allows us to easily find the expected value of a product of two independent random variables.

In the problem at hand, \({X}\) and \({Y}\) are independent, which means the calculation of the expected area of the rectangle simplifies as follows:

• Since \({E[X] = L}\) and \({E[Y] = L}\), the expected area is simply \({L^2}\).

Understanding independence helps us make these simplifications and accurately assess the expected outcomes in probabilistic situations.