let $X_1,X_2,\dots,X_n$ be random sample from the distribution $f(x)=e^{(\delta-x)}$ for $x\geq\delta$. show $\bar x$ is biased estimator and use it to find the unbiased estimator.
My solution:
To show its biased: $$E[\bar x]=E[\dfrac{1}{n}\sum_{i=1}^n x_i]=\dfrac{1}{n}\sum_{i=1}^n E[x_i]$$ $$E[x_i]=\int_{\delta}^\infty x_i e^{\delta-x_i}dx_i=1-\delta$$ Hence, $E[\bar x]=1-\delta \implies$ bias estimator.
Therefore, an unbiased estimator is $\hat \delta =1-\bar x$
Is my solution correct?