## 1. MLE of the Exponential Rate ## For $n > 1$, let $X_1, X_2, \ldots , X_n$ be i.i.d. exponential $(\lambda )$ variables. **a)** Let $\hat{\lambda}_n$ be the maximum likelihood estimate (MLE) of the parameter $\lambda$. Find $\hat{\lambda}_n$ in terms of the sample mean $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$. The subscript $n$ in $\bar{X}_n$ is there to remind us that we have the average of $n$ values. It doesn't refer to the $n$th sampled value $X_n$. **b)** Use facts about sums and linear transformations to find the distribution of $\bar{X}_n$ with little or no calculation. Recognize it as one of the famous ones and provide its name and parameters. Use it to find $E(\hat{\lambda}_n)$. **c)** Is $\hat{\lambda}_n$ an unbiased estimate of $\lambda$? If it is biased, does it overestimate on average, or does it underestimate? Is it asymptotically unbiased? That is, does $E(\hat{\lambda}_n)$ converge to $\lambda$ as $n \to \infty$