In a mixture of hydrogen and oxygen at the same temperature the hydrogen molecules will move faster than the oxygen molecules.
Temperature is a measure of average kinetic energy. Since the two types of gas molecules are at the same temperature they have the same average kinetic energy.
Kinetic Energy = 1/2 (mass)(velocity)^2
if 1/2 (mass of H2)(velocity of H2)^2 = 1/2 (mass of O2)(velocity of O2)^2 and the masses of H2 and O2 are 2 and 32 respectively, then:
(velocity of H2)^2 = 16(velocity of O2)^2, taking the square root of both sides we get:
velocity of H2 = 4 times the velocity of O2
This is the basis for Graham's Law of Diffusion/Effusion, which states that the ratio of speeds of two gases in inversely related to the square root of the ratio of their masses.