Examples of pairewise independent but not independent continuous random variables By considering the set $\{1,2,3,4\}$, one can easily come up with an example (attributed to S. Bernstein) of pairwise independent but not independent random variables. 
Counld anybody give an example with continuous random variables? 

An answer of mine on stats.SE gives essentially the same answer as the one given by vadim123.
Consider three standard
normal random variables $X,Y,Z$ whose joint probability
density function
$f_{X,Y,Z}(x,y,z)$ is not $\phi(x)\phi(y)\phi(z)$ where
$\phi(\cdot)$ is the standard normal density, but rather
$$f_{X,Y,Z}(x,y,z) = \begin{cases} 2\phi(x)\phi(y)\phi(z)
& ~~~~\text{if}~ x \geq 0, y\geq 0, z \geq 0,\\
& \text{or if}~ x < 0, y < 0, z \geq 0,\\
& \text{or if}~ x < 0, y\geq 0, z < 0,\\
& \text{or if}~ x \geq 0, y< 0, z < 0,\\
0 & \text{otherwise.}
\end{cases}\tag{1}$$
We can calculate the joint density of any pair of the random variables,
(say $X$ and $Z$) by integrating out the joint density with respect to
the unwanted variable, that is,
$$f_{X,Z}(x,z) = \int_{-\infty}^\infty f_{X,Y,Z}(x,y,z)\,\mathrm dy.
\tag{2}$$


*

*If $x \geq 0, z \geq 0$ or if $x < 0, z < 0$, then
$f_{X,Y,Z}(x,y,z) = \begin{cases} 2\phi(x)\phi(y)\phi(z), & y \geq 0,\\
0, & y < 0,\end{cases}$ and so $(2)$ reduces to
$$f_{X,Z}(x,z) = \phi(x)\phi(z)\int_{0}^\infty 2\phi(y)\,\mathrm dy = 
\phi(x)\phi(z).
\tag{3}$$

*If $x \geq 0, z < 0$ or if $x < 0, z \geq 0$, then
$f_{X,Y,Z}(x,y,z) = \begin{cases} 2\phi(x)\phi(y)\phi(z), & y < 0,\\
0, & y \geq 0,\end{cases}$ and so $(2)$ reduces to
$$f_{X,Z}(x,z) = \phi(x)\phi(z)\int_{-\infty}^0 2\phi(y)\,\mathrm dy = 
\phi(x)\phi(z).
\tag{4}$$
In short, $(3)$ and $(4)$ show that $f_{X,Z}(x,z) = \phi(x)\phi(z)$ for all 
$x, z \in (-\infty,\infty)$ and so $X$ and $Z$ are
(pairwise) independent standard normal random variables. Similar
calculations (left as an exercise for the bemused
reader) show that $X$ and $Y$ are
(pairwise) independent standard normal random variables, and
$Y$ and $Z$ also are
(pairwise) independent standard normal random variables.  But
$X,Y,Z$ are not mutually independent normal random variables.
Indeed, their joint density $f_{X,Y,Z}(x,y,z)$
does not equal the product $\phi(x)\phi(y)\phi(z)$ of
their marginal densities for any choice of 
$x, y, z \in (-\infty,\infty)$
