How can a proton be converted to a neutron via positron emission and yet gain mass? The mass of a neutron is greater than mass of a proton so how is it possible in positron emission for a proton to form a neutron and a positron?
 A: Because the mass of a nucleus isn't just the sum of its parts.  Positron emission obeys the nuclear mass-energy balance like all other nuclear reactions.  The mass deficit is the energy of the reaction.
In other words, the reaction still decreases the total mass (reactants versus products).  Your observation (that a neutron is heavier than a proton) tells us that positron emission can't occur for a proton in isolation - and this is true.
Binding energy is one way to represent the fact that a nucleus's weight isn't just the sum of its parts.  So another way to answer your question is to say that the change in binding energy is more than enough to make up for the increase in mass due to changing from a proton to a neutron.  So that transition (p to n) causes an increase in nucleon mass, but the binding energy causes a decrease in mass (for the decay).  The fact that positron emission happens evidences the fact that the change in binding energy is greater.
A: In addition to Alan's notes about keeping track of total energy in a nuclear context, it is also important to keep track of the neutrino. 
Even free protons can be converted to neutrons (a process called "inverse beta decay") if there is an incident anti-neutrino of sufficient energy: 
$$p + \bar{\nu}_e \to n + e^+ \,.$$
This is the detection mechanism that is used for reactor anti-neutrinos oscillation measurements (the delayed coincidence between the positron annihilation and the subsequent capture of the free neutron is very clean), and requires a anti-neutrino of 1.8 MeV or more to proceed.
The pure decay is
$$ p \to n + e^+ + \nu_e \,,$$
and requires a little more energy that you would naively expect from the masses of the proton, neutron and positron (and, of course, can only proceed in a context where there is energy to be had doing it, such as in some heavy nuclei).
A: when anti neutrino collide with protons (process known as"Reverse BETA DECAY") they produce neutrons and electrons
