Puzzles on continuous probability
Example 1 What is the maximum possible variance of a random variable that takes the values in the set \([-1,1]\)?
Solution.
Intuitively speaking, the variance of a mass distribution is the probability weighted average of the sum of the squared distances from the mean. It is maximized if the masses are placed far away from the center of gravity. So, if we place two point masses with probability weights \(1/2\) at \((-1,0)\) and \((1,0)\), the variance would be maximal. Hence, the maximum variance equals \(1\).
Example 2 What is the maximum possible variance of a random variable that takes the values in the set \([0,1]\)?
Solution.
Again, we can have point masses with probability weights \(1/2\) at \(X=0\) and \(X=1\). The center of mass or expectation of this distribution \(EX = 1/2\). So, the maximal variance would be \((1/2)(1/2)^2 + (1/2)(1/2)^2=1/4\).
Example 3 Let \(X\) be a random variable such that \(P(X \ne 0) > 0\). Suppose that for some real numbers \(a\) and \(b\), the random variables \(aX\) and \(bX\) have the same distribution. Is it true that \(a = b\)? What if we also assume that \(a\) and \(b\) are both positive.
Solution.
If \(aX\) and \(bX\) have the same mass distribution, they have the same expectation and variance.
Hence, \(\mathbb{E}[aX] = \mathbb{E}[bX]\). So, \(a\mathbb{E}[X] = b \mathbb{E}[X]\). Consequently, \((a-b)\mathbb{E}[X] = 0\). If \(a \neq b\), then \(\mathbb{E}[X] = 0\). For example, consider the \(U \sim U[-1,1]\) random variable. Both \(U\) and \(-U\) have the same distribution. Also, their second moments must match. So, \(\mathbb{E}[aX\)