A mixed unequal-probability/SRSWOR design — the sample mean is biased, but an unbiased estimator still exists

In a survey of finite population, the following method is applied to obtain a sample of size $n$: (i) In the 1st draw, the $i$th unit is selected with probability $P_i>0, i=1,\dots,N, \sum P_i=1$ and $P_i\neq1/N$ for some $i$. (ii) The remaining $(n-1)$ units are obtained by SRSWOR method from remaining $N-1$ units of the population. Let $\overline Y$ denote the sample mean, then: