Expected value of the area of square

So basically, the expected value of the area of a square with side length X is not equal to the square of the expected value of X, which is E[X]^2. This is because the expected value of X^2 takes into account the variability in X, while E[X]^2 does not. Therefore, the two are not equivalent and cannot be interchanged. In summary, the expected value of the area of a square with side length X is not equal to the square of the expected value of X due to the fact that E[X]^2 does not take into account the variability in X, while E[X^2] does.
  • #1
Glass
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Homework Statement


Given a square with side lengths X, where X is a random variable with some probability density function (the actual pdf is not important for my question). Why isn't the expected value of the area = E[X]^2 = E[X^2]?


Homework Equations





The Attempt at a Solution


Intuitively I would think, if I can find the expected value of one of the sides, I can get the expected value of the area by squaring it.
On the other hand, I am well aware that E[X]^2 != E[X^2] in the general case, and this is indeed a general case (with some geometric interpretation).
 
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  • #2
Wait nevermind, I got it.
 

1. What is the expected value of the area of a square?

The expected value of the area of a square is the average value of all possible outcomes when the square is randomly selected. It is a measure of the central tendency of the area of a square.

2. How is the expected value of the area of a square calculated?

The expected value of the area of a square can be calculated by multiplying the length of one side of the square by itself (squared) and then multiplying that result by the probability of selecting that square. This process is repeated for each possible outcome and the results are added together.

3. Why is the expected value of the area of a square important?

The expected value of the area of a square is important because it provides a mathematical representation of what we can expect to happen on average when selecting a square at random. It can also be used to make predictions and inform decision-making in various fields such as statistics, economics, and game theory.

4. Can the expected value of the area of a square be negative?

No, the expected value of the area of a square cannot be negative. The area of a square is always a positive value, and the expected value is a measure of the average of all possible outcomes. Therefore, it cannot be negative.

5. How does the expected value of the area of a square relate to the concept of probability?

The expected value of the area of a square is closely related to the concept of probability. It is calculated by multiplying the area of each possible outcome by its corresponding probability. In other words, it is a weighted average of all possible outcomes, with the weights being the probabilities of those outcomes occurring. This relationship allows us to use expected value as a tool for understanding and predicting outcomes in probability scenarios.

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