Calculating the Expected Area of a Target Square within a Unit Square

In summary, the problem involves finding the average size of a target square of size LXL = p^2 < 1 in a unit square. The center of the target square is equally likely to be anywhere in the unit square. The problem is divided into three cases: when the target square is completely inside the unit square, when half of the target square is inside the unit square, and when a quarter of the target square is inside the unit square. The expectation integral of p^2 is calculated, but there is confusion about the probability density function and the limits of integration for each case. Help in determining these is needed.
  • #1
purplebird
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Homework Statement



Given a unit square and a 'target' square of size LXL = p^2 < 1 in the unit square. The center of target square in equally likely to be anywhere in the unit square. What is the average size of the target square as a function of p^2.

This is the problem and I have included a jpeg illustration of the problem. Any help would be greatly appreciated.

My attempt at the problem :

I divided the problem into three cases : one where the target square is completely in the unit square : one where half the target square is in the unit square and one where quarter of the target square is in the unit square. I then calculated the expectation integral of p^2 but I am confused as to what the probability density function is and what the limits of integral are for each of the cases.
 

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I have tried finding out the probability density functions and limits of integral online and also by myself but I am not able to find them. Any help in this regard would be appreciated.
 

What is the expected value of area?

The expected value of area is a statistical concept that represents the average area that would be obtained if an experiment were repeated an infinite number of times. It is calculated by multiplying the area of each possible outcome by its probability of occurring and summing all the values.

Why is the expected value of area important?

The expected value of area is important because it provides a way to quantify the average outcome of a random experiment. It is used in various fields such as economics, finance, and engineering to make informed decisions and predictions based on probabilities.

How is the expected value of area calculated?

The expected value of area is calculated by multiplying the area of each possible outcome by its probability of occurring and then summing all the values. This can be represented mathematically as E(A) = Σ(A * P(A)), where A represents the area and P(A) represents the probability of that area occurring.

What are some real-life applications of the expected value of area?

The expected value of area has many real-life applications. For example, it can be used to determine the average profit of a business venture, the expected return on an investment, or the expected loss in a game of chance. It is also used in risk analysis to evaluate potential outcomes and make decisions based on probabilities.

What are the limitations of the expected value of area?

While the expected value of area is a useful tool for making decisions based on probabilities, it does have its limitations. It assumes that all outcomes are equally likely, which may not always be the case in real-world scenarios. It also does not take into account potential extreme or rare outcomes, which can greatly affect the actual outcome of an experiment.

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