In this paper, we focus on the degree of the greatest common divisor (
gcd
) of random polynomials over
F
q
. Here,
F
q
is the finite field with
q
elements. Firstly, we compute the probability distribution of the degree of the
gcd
of random and monic polynomials with fixed degree over
F
q
. Then, we consider the waiting time of the sequence of the degree of
gcd
functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type
gcd
, we investigate the probability distribution of the number of rational (i.e., in
F
q
) roots (counted with multiplicity) of random and monic polynomials with fixed degree over
F
q
.