Assuming we observe events that follow a particular distribution, like exponential bayesian-inference-continuous-distributions, how do we determine $\lambda$, its parameter?

This question is what motivates statistics. In the real world, $\lambda$ is hidden from us. We see only Z, and must go backwards to try and determine $\lambda$. The problem is difficult because there is no one-to-one mapping from Z to $\lambda$. Many different methods have been created to solve the problem of estimating $\lambda$, but since it is never actually observed, no one can say for certain which method is best!

Bayesian inference is concerned with beliefs about what $\lambda$ might be. Rather than try to guess it exactly, we can only talk about what $\lambda$ is likely to be by assigning a probability distribution to $\lambda$

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