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As I understand a random variable represents all possible outcomes of an experiment with their associated probabilities. Why $X^2$ is understood as squaring outcomes of experiments instead of as multiplying the results of two identical experiments?

Let's X be a throw of a die. I've seen examples where expected value of $X^2$ is calculated as: $$ 1^2\dfrac{1}{6} + 2^2\dfrac{1}{6} + 3^2\dfrac{1}{6} + 4^2\dfrac{1}{6} + 5^2\dfrac{1}{6} + 6^2\dfrac{1}{6} = 15.16 $$

So here we just squaring the outcome of a single experiment. But in math square can be replaced by multiplication. Like: $X\times X$. And for multiplication of two random variable we probably need to construct a product distribution. As I understand it should be something like throwing 2 dice, multiplying the results, and then calculating the average of these values. So as both X are independent in this case expected value can be found as $E[XY]=E[X]E[Y]$. I.e. $3.5 *3.5 = 12.25$

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    $\begingroup$ X and X are not independent. $\endgroup$ – Michael M Jun 29 '14 at 9:32
  • $\begingroup$ X is an experiment of throwing a dice. Why two experiments like that are not independent? $\endgroup$ – Vasili Jun 29 '14 at 9:35
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    $\begingroup$ The random variables representing outcomes from rolling two die rolling experiments are independent, but that would not be written as $X\times X$. It would be $X_1 \times X_2$ - the product of the two random variables. $X\times X$ represents the square of one random variable, not the product of two different ones. $\endgroup$ – Glen_b Jun 29 '14 at 9:49
  • $\begingroup$ Make sense, thanks! $X_1 \times X_2 $ fills my gap in the notation. $\endgroup$ – Vasili Jun 29 '14 at 9:59
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Your example correctly shows that $X^2$ and $X \cdot Y$ are not distributed equally. So where is the difference between the two settings?

A random variable is a fixed function that quantifies each possible outcome of a prespecified experiment.

In your example, there are two such experiments:

  1. The first one is "Roll the first die" quantified by $X = $ number of
    dots on the first die,
  2. while the second one is "Roll the second die (independent of the first)"
    quantified by $Y = $ number of dots on the second die.

Although the distributions of $X$ and $Y$ are the same (they take the values 1 to 6 each with probability $1/6$), $X$ and $Y$ are not equal since the first roll does not always equal the second roll.

The random variable $X^2$ on the other hand belongs only to the first experiment "Roll the first die" and is defined as $X^2 = $ squared number of dots on the first die. It differs from $Y$ in just looking twice at the same roll instead of looking at two different rolls.

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