When do mean value may be equal to some percentile? Let’s say 3rd percentile.

Question

I started reading about statistics recently and saw that this question was asked several times during interviews.

Answer 1

A symmetrical random variable has mean $\mu$ equal to median $\eta,$ provided that the median exists.The median is the 50th percentile. For example, the standard normal distribution has $\mu = \eta = 0,$ but Student's t distribution with one degree of freedom (sometimes called standard Cauchy) is symmetrical about $0,$ so $\eta = 0.$ b However, $\mu$ does not exist.

For skewed distributions (such as exponential, gamma, Weibull, Pareto, and so on) the mean (when it exists) may occur at various percentiles. For example, the standard exponential distribution has $\mu = 1$ and $\eta \approx 0.6931.$ But if $X \sim \mathsf{Exp}(1),$ then $P(X \le 1) \approx 0.6321$ so that the mean $\mu$ is located at about the 63rd percentile (or at quantile 0.6321206). Computations from R statistical software below:

qexp(.5)
## 0.6931472
pexp(1)
## 0.6321206

The histogram below shows the density function of the standard exponential distribution. The solid vertical red line is at $\mu = 1;$ the broken blue line is at $\eta \approx 0.6931.$ About 63% of the area lies beneath the curve between $0$ and $\mu = 1.$