published an idea
is similar to Cotner’s Observation which I will formally state below.
If you see a tall person from a distance, you tend to expect that the tall person is male. In fact, the taller the person is, the more likely it is that the person is male. If you plot histograms of people’s heights, you will find that this intuition is correct. Cotner once pointed out to me that this intuition is false for many non-Gaussian Distributions.
Cotner’s Observation:$\ $
Consider two random variables, $A$ and $B$ with $P(B>A) > 0.5$ and a random variable $X$ which is always equal to $A$ or $B$. Suppose we flip a fair coin and if the coin is heads we assign $X$ to $A$ and if it is tails we let $X$ equal $B$. Now if $X$ is large, we tend to think the coin was probably tails because $B$ is usually larger than $A$. Furthermore, the larger $X$ is the more we think that the coin was probably tails. This intuition is right for Gaussian Distributions, but it is wrong for many other distributions. Let $p(x)$ be the probability that the coin was heads given that $X = x$
$$p(x) := P( X=A | X=x).$$
- If $A$ and $B$ have Gaussian Distributions with equal standard deviations, then $p(x)$ is decreasing.
- If $A$ and $B$ have Laplace Distributions with equal standard deviations, then $p(x)$ is becomes constant for all $x > E[B]$ and $p(x) < 0.5$ when $x > E[B]$.
- If $A$ and $B$ have Student T-Distributions with equal standard deviations and equal degrees of freedom, then $p(x)$ will actually increase if $x$ is large enough and $p(x)$ approaches 0.5 as $x$ goes to infinity.
Many distributions have a fat tails like the Student T Distributions.
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