Intuitive examples of distributions where the mean is far from the most probable outcome.

Question

In general, the peak of a probability distribution does not coincide with its mean unless it is symmetric. I recently explained this to a student who made the mistake of identifying the two, but I didn't have off the top of my head a nice really obvious and intuitive example of an "everyday" distribution which most people would be familiar with in which the mean and most probable values are very distinct to drive the point home. This question is mostly just asked out of curiosity if anyone has some nice examples for pedagogical purposes.

Edit:

For anyone interested, the original problem was to look at the probability distribution of the current through a diode $$I(V) = I_0(e^{eV/kT}-1)$$ subject to a gaussian distributed "noisy" potential (e.g. induced by thermal effects at finite temperature, with zero mean). Depending on the temperature, the most probable current can be negative while the average current is positive.

Answer 1

Anything on an exponential distribution. We can add anything on a Pareto distribution, a log-normal distribution and many others. All of these are long-tailed distributions.

The distribution of wealth in the economy.

The brightness of stars.

The sales of books on Amazon.

The distribution of mailbox numbers in a city.

Answer 2

In skewed examples, the mean is dragged in the direction of the skew (tail).

• A random variable that takes value $1$ with probability $3/4$ and value $100$ with probability $1/4$ has mean $25.75$, compared to the most probable outcome $1$.
• If $\lambda$ is not an integer, a Poisson distribution with mean $\lambda$ has most probable outcome $\lfloor \lambda \rfloor$.
• If you allow continuous distributions, and replace "most probable outcome" with "mode (maximizer of PDF)", then the exponential distribution with rate $\lambda$ has mean $1/\lambda$ while the mode is $0$.

Answer 3

If you break a wooden stick of length 1 randomly (selecting the breaking point with uniform probability) and denote the length of the longer and shorter part by X,Y, respectively, then $$\mathbb{E}\frac{X}{Y}=\infty,$$ while the peak of the density is achieved by 1. Could be a decent exercise for a student to actually verify that.

Answer 4

Anything that is "very bimodal". E.g. how about $X= $ number of Heads in a single coin flip? The "peaks" are $0$ and $1$ and the mean is $1/2$.

If you prefer something continuous, just do $Y = X+ N(0, 0.00001)$. You get two very narrow Gaussian peaks and their tails' overlap is negligible.