Can water be produced by mixing hydrogen gas and oxygen?
Indeed, it's called combustion! When you burn hydrogen and oxygen gas, the reaction is:

$$\ce{2H2_{(g)} + O2_{(g)} -&gt; 2H2O_{(g)} + energy}$$

Basically, you could see the former U.S. space shuttle as a large machine that turns hydrogen and oxygen into water. (Note: the space shuttle used liquid hydrogen and oxygen, but otherwise the principle and formula is exactly the same)

Since the Gibbs free energy of the forward reaction is negative ($-242\textrm{ kJ/mol}$), you don't need to add energy to the system, the reaction is spontaneous (mind you, you generally need a spark to ignite the combustion because there's a kinetic barrier to the reaction called the energy of activation).

The reverse reaction, water to hydrogen and oxygen, has a positive Gibbs free energy, so you need to add energy in the form of electricity to the system to make the reaction happen.