  We construct a class of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
generated by pairs of polynomials, study their algebraic structures, and obtain
the generator matrix of any code in the class. Using a probabilistic method, we
prove that, for any positive real number $\delta<1/3$ such that the entropy at
$3\delta/2$ is less than $1/2$, the probability that the relative minimal
distance of a random code in the class is greater than $\delta$ is almost $1$;
and the probability that the rate of the random code equals to $1/3$ is also
almost $1$. As an obvious consequence, the $\mathbb{Z}_2\mathbb{Z}_4$-additive
cyclic codes are asymptotically good.
