In this paper, we investigate the outage probability and power allocation for the two-way decode-and-forward (DF) relay networks with relay selection.~Specially, we consider independent but not necessarily identical distributed Rayleigh fading channels. Firstly, we derive an exact closed form outage probability expression. To shed light on the relation between the outage probability and the power allocation factor, an upper bound for the outage probability is derived, too. We then propose a power allocation scheme in the sense of minimizing this upper bound. Monte Carlo simulations are conducted to show that the derived outage probability expression excellently matches simulation results, and our proposed power allocation scheme performs effectively.
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