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Find the method of moments estimate for $\lambda$ if a random sample of size $n$ is taken from the exponential pdf,

$$f_Y(y_i;\lambda)= \lambda e^{-\lambda y} \;, \quad y \ge 0$$

$$E[Y] = \int_{0}^{\infty}y\lambda e^{-y}dy \\ = \lambda \int_{0}^{\infty}ye^{-\lambda y} dy \\ = -y\frac{e^{-\lambda y}}{\lambda}\bigg\rvert_{0}^{\infty} - \int_{0}^{\infty}e^{-\lambda y}dy \\ =\bigg[\frac{e^{-\lambda y}}{\lambda}\bigg]\bigg\rvert_{0}^{\infty} \\ E[Y] = \frac{1}{\lambda} \\ $$Now solve for $\bar{y}$

$$E[Y] = \frac{1}{n}\sum_\limits{i=1}^{n} y_i \\ \bar{y} = \frac{1}{\lambda} \\ \lambda = \frac{1}{\bar{y}} $$

Implies that $\hat{\lambda}=\frac{1}{\bar{y}}$

Did I get this one? I have not got the answer for this one in the book.

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