We have $Y_{1}, Y_{2}, Y_{3}$ a random sample from an exponential distribution with the density function $ f(y) = \left\{ \begin{array}{ll} (1/\theta)\mathrm{e}^{-y/\theta} & y \gt 0 \\ 0 & elsewhere. \end{array} \right.$
I'm suppose to find which of the following estimators are unbiased: $\hat{\theta_{1}} = Y_{1}, \hat{\theta_{2}} = (Y_{1} + Y_{2}) / 2, \hat{\theta_{3}} = (Y_{1} + 2Y_{2})/3, \hat{\theta_{4}} = \bar{Y}$.
As far as I can tell none of these estimators are unbiased. For example
$ E(\hat{\theta_{1}}) \\ = E(Y_{1}) \\ = Y_{1}\int_0^\infty (1/\theta)\mathrm{e}^{-y/\theta}\,\mathrm{d}y \\ = \left.Y_{1}(-\mathrm{e}^{y/\theta}) \right|_0^\infty \\ = Y_1(0 + 1) = Y_1 $
and
$E(\hat{\theta_{4}}) \\ = E(\bar{Y}) \\ = \int_0^\infty (1/\theta^2)\mathrm{e}^{-2y/\theta}\,\mathrm{d}y \\ = \left.(1/2\theta)(-\mathrm{e}^{-2y/\theta}) \right|_0^\infty \\ = (1/2\theta)(0 + 1) = 1/2\theta$
So it looks like none of these are unbiased. I imagine the problem exists because one of $\hat{\theta_{1}}, \hat{\theta_{2}}, \hat{\theta_{3}}, \hat{\theta_{4}}$ is unbiased.
What am I doing wrong?