Functional relationship misleads statistical conjecture -how to explain?

Question

This is an educational issue for me (I am not a Statistics instructor, but colleagues tend to turn to me for some first-line help with the subject), but I believe it bears to anyone having to present statistical results in a professional capacity in the industry.

I have two non-negative random variables, functionally linked:

$$X = g(Y), \;\;g'(Y) <0$$

So for higher values of $Y$, the value of $X$ gets lower. Assume that $Y$ has a unimodal distribution skewed to the right. Informally one then thinks that "the high-end values of $Y$ are less probable, so if the distribution is stable over time, we will tend to see these higher values less often". And here comes the powerful logical trap: "So, since high values of $Y$ occur less often, it must be the case that the low-end values of $X$ occur less often, due to the relation linking the two variables, and so we should expect that the distribution of $X$ is skewed to the left".

Which, simply, it is not the case, at least not generally. Consider the simplest example: Let $Y$ follow a log-normal distribution, so we have

$$\ln Y \sim N(\mu, \sigma^2)$$

and let $$X = Y^{-a}$$

Then

$$\ln X= -a\ln Y \implies \ln X \sim N(-a\mu,a^2\sigma^2)$$

So $X$ will also follow a log-normal distribution, which is also skewed to the right... meaning, that even though $X$ is inversely related to $Y$, their distributions will be qualitatively similar -in both cases, the high-end values will appear "relatively less often".

Certainly, the mathematics are indisputable, but I find this very hard to explain to people. To be honest I don't think I have ever managed to really help anyone understanding it, since in most cases I remember ending with something like "unlike conventional impression, statistics are counter-intuitive more often than not" -which hardly contributes to an understanding.

This is my question then: How could one understand, and communicate the resolution of the apparent "logical conflict" between the functional relationship of these two random variables, and the fact that their statistical distributions seem to "negate" it?

Answer 1

It may help your audience to begin with the fact that $g$ maps quantiles to one another exactly. The trick, then, is to explain how quantiles fail to correspond with visual space on a plot.

Your example is great for ease of algebra, but maybe a simple discrete example would give your audience a useful picture. Suppose $Y$ is 1/5, 1, or 5 with probabilities 0.1, 0.8, and 0.1, respectively. Suppose $g$ is $y \mapsto 1/y$, so it maps 1/5 to 5, 1 to 1, and 5 to 1/5. Then $X$ has the same distribution as $Y$. This is essentially your example with $a = 1$, but it makes visible the phenomenon causing the paradox. In this example, the 0.8 probability mass at 1 stays "low" the whole time. The cause is that $g$ stretches out a region of upper quantiles, tricking our eyes into adopting a strict definition of "high values."