My friend was asked this as an interview question, and was confused, as surely there is no maximum and minimum just the product and what the expectation of that is. The three numbers x,y,z are $\in \left[ 0,1\right] $
$\int ^{1}_{0}\int ^{1}_{0}\int ^{1}_{0}xyzdxdydz = \left( \dfrac {1}{2}\right) ^{3}$.
Have I missed something here.
On
Unless you have the precise wording of the question, we can only speculate about its intended meaning. One such speculation is that they may have meant that the three possible products of pairs of the three numbers are formed, and we want the expectations of the minimum and maximum products. This would be
$$ 6\int_0^1\mathrm dx\int_0^x\mathrm dy\int_0^y\mathrm dz\;yz=3\int_0^1\mathrm dx\int_0^x\mathrm dy\;y^3=\frac34\int_0^1\mathrm dx\;x^4=\frac3{20} $$
for the minimum and
$$ 6\int_0^1\mathrm dx\int_0^x\mathrm dy\int_0^y\mathrm dz\;xy=6\int_0^1\mathrm dx\int_0^x\mathrm dy\;xy^2=2\int_0^1\mathrm dx\;x^4=\frac25 $$
for the maximum. Of course we can also do the same for the median product:
$$ 6\int_0^1\mathrm dx\int_0^x\mathrm dy\int_0^y\mathrm dz\;xz=3\int_0^1\mathrm dx\int_0^x\mathrm dy\;xy^2=\int_0^1\mathrm dx\;x^4=\frac15\;. $$
The average is $\frac13\left(\frac3{20}+\frac25+\frac15\right)=\frac14$, as it should be for the product of two random variables independently uniformly distributed on $[0,1]$, as this is simply the square of the mean $\frac12$.
The way I understand this is:
$\int ^{1}_{0}\int ^{1}_{0}\int ^{1}_{0}xyzdxdydz = \int ^{1}_{0}\int ^{1}_{0}\int ^{1}_{0}0\cdot 0\cdot 0 \ dxdydz = 0^3 = 0$ is the minimum and$\int ^{1}_{0}\int ^{1}_{0}\int ^{1}_{0}1\cdot 1\cdot 1 \ dxdydz = 1^3 = 1$ is the maximum.