Let $X_1, X_2, \ldots , X_n$ be a random sample from a distribution having probability density function (pdf)
$$f(x\mid \theta) = \theta e^{−\theta x},\quad \theta > 0, x > 0$$
Derive the likelihood function for $\theta$, maximum likelihood estimator (MLE) of $\theta$ and its asymptotic distribution.
I don't understand this topic very well, how can I derive the likelihood function for $\theta$?
Hint:
Likelihood function for $\theta$ is just the probability density function of your sample, but regarded as a function of the parameter $\theta$ instead of the observations. $L(\theta|\mathbf{x})=f(\mathbf{x}|\theta)$.
In this case, since $X_i$ are independent, the probability density function of the sample is the product of each individual pdf. Then you maximize this function with respect to $\theta$ to get the MLE estimate of $\theta$.