Method of moments $f(x;\lambda) = \lambda e^{(-\lambda x)}$ for $x > 0$

Question

Find the method of moments estimator of the parameter $\theta$ if you have a random sample of size $n$ form the following distribution:

$$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & \text{for } x > 0 \\ 0& \text{otherwise } \end{cases}$$

So this is what I got so far

$$E(x) = \int_{0}^{\infty} x \lambda e^{-\lambda x} dx$$

So here I have a decision to make, whether to do a Laplace transform (might not be possible) or integration by part! So which way is better?

Answer 1

Correction:

$$\mathbb{E}(X)=\int_0^\color{red}\infty x\lambda \exp(-\lambda x) \, dx$$

To evaluate the above Any method that works is fine. For me, I would just look up the final answer directly since expectation of exponential distribution is well known.

Equate $$\frac{\sum_{i=1}^nx_i}{n} = \mathbb{E}(X)$$ to find the estimate $\hat{\lambda}$.