Solved – Variance of estimator(exponential distribution)

estimatorsmethod of momentsself-studyvariance

I have exponential distributed data $Exp(\lambda)$ with sample n = 50. Also, The sample mean = 2.17. I need to find the estimator of parameter $\lambda$ by the method of moments and to build 95% confidence interval.

We know, that by the method of moments we got the estimation the parameter $\lambda$ = $1/\bar X$.

Now for confidence interval of my estimator i must calculate the variance of estimator: $Var(\hat \lambda)$ = $Var(1/\bar X)$

But now i am not sure what to do. My attempt: $1/Var(\bar X) = 1/Var(\sum (1/n) *X_i)$ and by the propery o variance i got the result $\lambda^2 * n$.

But i doubt, that is correct result. Can you please help me?
many thanks

Best Answer

$$Var\left(\dfrac{1}{\bar{X}} \right) \ne \dfrac{1}{Var(\bar{X})}.$$

So what you have is indeed incorrect. To solve the problem, go through the following steps:

  1. What is the distribution of $\sum_{i=1}^{n} X_i$?
  2. What is the distribution of $\bar{X}$?
  3. What is the distribution of $Y = 1/\bar{X}$?
  4. What is $Var(Y)$?
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