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Without more knowledge on $X$, we cannot say a lot. However, if it is IID with mean $\mu$ and variance $\sigma^2$, the sample mean $\hat{\mu} = \frac{1}{N}\left(\sum_{n=0}^{N-1}x_n\right)$ is unbiased, since:
$$E[\hat{\mu}] = E\left(\frac{1}{N}\sum_{n=0}^{N-1}x_n\right) = \frac{1}{N}\sum_{n=0}^{N-1}E\left(x_n\right) = N\mu/N=\mu\,.$$
And its variance goes to ...
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A sine wave is cyclo-stationary so is not the best waveform to use for estimating delay between transmit and received signals. It will correlate whenever the delay causes the signal to be an additional $2\pi$ in phase offset. Consider using waveforms with better autocorrelation properties for this purpose such as pseudo-random noise sequences or other ...
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