Suppose we have a random sample $(X_{1}, \dots,X_{n})$, where $X_{i}$ follows an exponential distribution with parameter $\lambda$. Hence, we have the following:
$$F(x) = 1 - \exp(-\lambda x)$$
$$\mathbb{E}(X_{i}) = \dfrac{1}{\lambda}$$
$$\text{Var}(X_{i}) = \dfrac{1}{\lambda^{2}}$$
I know that the MLE estimator is $\hat{\lambda} = \dfrac{n}{\sum_{i=1}^{n}X_{i}}$.
I am interested in estimating the CDF at the point $x = 1$, hence what I have is
$$\hat{F(1)} = 1 - \exp(- \hat{\lambda})$$
I want to discover if it is biased or not:
\begin{align*} \mathbb{E}\left( \hat{F}(1) \right) &= 1 - \mathbb{E}(\exp(- \hat{\lambda})) \end{align*}
Any clues how to proceed from here?
The question is only to detect if the estimator is biased or not. They do not ask you to explicitly calculate the expectation of your estimator. Thus the solution is very easy: using Jensen's inequality you get
$$\mathbb{E}[\hat{F(1)}]=\mathbb{E}\Bigg[1-e^{-\frac{1}{\overline{X}_n}}\Bigg]\ne 1-e^{-\frac{1}{\mathbb{E}[\overline{X}_n]}}=1-e^{-\lambda}$$
Thus your estimator is biased
EDIT: I supposed it was easy also to calculate the needed expectation explicitly passing by Gamma function but I was wrong...the integral can be solved numerically... thus the solution (IMHO) is the one I showed