$a$ and $b$ are different known positive constants and with an unknown parameter $p\in(0,1)$, a random variable $X$ follows $$g_p(x)=p\frac{a^x}{x!}e^{-a}+(1-p)\frac{b^x}{x!}e^{-b}\quad(x=0,1,2...)$$
I need to get an unbiased estimator of p.
What I have tried
I have got its expected value and variance $$E(X)=pa+(1-p)b\\\ V(X)=a^2p+(1-p)b^2$$
I do not know how I can get an unbiased estimator of the parameter in PDF. Can anyone help me?