Find each of the following product:

To multiply algebraic expressions, we use commutative and associative laws along with the the law of indices, that is, $a^{m} \times a^{n}=a^{m+n}$.

We have:

$\frac{1}{4} x y \times \frac{2}{3} x^{2} y z^{2}$

$=\left(\frac{1}{4} \times \frac{2}{3}\right) \times\left(x \times x^{2}\right) \times(y \times y) \times z^{2}$

$=\left(\frac{1}{4} \times \frac{2}{3}\right) \times\left(x^{1+2}\right) \times\left(y^{1+1}\right) \times z^{2}$

$=\frac{1}{6} x^{3} y^{2} z^{2}$

Thus, the answer is $\frac{1}{6} x^{3} y^{2} z^{2}$.