$\theta_1(X)$ is an estimator, where $\mathbb{E}_{\theta}[\theta_1(X)]=\frac{n-1}{n}\theta$
Is $\theta_2(X)=\frac{n}{n-1} \theta_1(X) $ an unbiased estimator of $\theta$
I know that $\theta_1(X)$ is a biased estimator, since it’s mean is a function of $\theta$, but unaware of how to check if $\theta_2(X)$ is unbiased (unless there is a way of calculating the mean?
An estimator $w_\theta(\boldsymbol X)$ of some parameter $\theta$ is unbiased if it satisfies the property $$\operatorname{E}[w_\theta(\boldsymbol X)] = \theta.$$ Therefore, if $$\operatorname{E}[w_\theta(\boldsymbol X)] = c \theta$$ for some nonzero constant $c \ne 1$ with respect to $\theta$, then the modified estimator $$\frac{1}{c} w_\theta(\boldsymbol X)$$ is unbiased for $\theta$, since $$\operatorname{E}\left[\frac{1}{c} w_\theta(\boldsymbol X)\right] = \frac{1}{c}\operatorname{E}[w_\theta(\boldsymbol X)] = \frac{1}{c} \cdot c\theta = \theta.$$ The only requirement is that $c$ is not a function of $\theta$, and $c \ne 0$.