Graphical explanation of independent and uncorrelated RVs The concept of mean independence is often used in statistics with two implications:

*

*strong assumption of independent random variables (X1⊥X2)

*weak assumption of uncorrelated random variables (Cov(X1,X2)=0)

Is it possible to provide some kind of graphical explanation for these two types of mean independence.
 A: For randomly sampled normal observations $X_1, X_1, \dots, X_n,$ one can show that the sample mean $A_x = \bar X$ and sample variance $S_x^2$ are independent random variables. This is true only for normal data.
In contrast, by symmetry, a random sample from $\mathsf{Beta}(.1, .1)$ has uncorrelated $A_y = \bar Y$ and $S_y^2,$ but it is easy to see that the sample mean and variance are not independent random variables.
(You asked for a graphical illustratin, so I will use sample standard deviations instead of sample variances.)
set.seed(123)
m = 50000;  n = 5

x = rnorm(m*n)
DTA.x = matrix(x, ncol=n)
a.x = rowMeans(DTA.x)
s.x = apply(DTA.x, 1, sd)
cor(a.x, s.x)
[1] 0.001337359   # aprx 0

y = rbeta(m*n, .1, .1)
DTA.y = matrix(y, ncol=n)
a.y = rowMeans(DTA.y)
s.y = apply(DTA.y, 1, sd)
cor(a.y, s.y)
[1] 0.002938279   # aprx 0
mean(a.y < .2)
[1] 0.05938       # P(A.y < .2) > 0
mean(s.y > .45)
[1] 0.52144       # P(S.y > .45) > 0
mean(a.y<.2 & s.y>.45) 
[1] 0             # P(A.y<.2, S.y>.45) = 0
                  # Independence fails

In the beta plot in the the right-hand panel of the figure below,
the vertical strip to the left of 0.2 and the horizontal strip
to the above 0.45 each have positive probability, but their
intersection does not, which shows that $A_y = \bar Y$ and
$S_y$ are not independent random variables.

R code for figure:
par(mfrow = c(1,2))
plot(a.x, s.x, pch=".", main = "Normal Data")
plot(a.y, s.y, pch=".", main = "BETA(.1,.1) Data")
 abline(v=.2, col="red");  abline(h=.45, col="red")
par(mfrow = c(1,1))

Note: The original simulated beta data lie in a 5-dimensional hypercube, mostly
near faces, edges and vertices. In the figure
at right above, these data have been
transforms ('squashes') into two dimensions, but some evidence of the
edges and vertices of the hypercube remains.
