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This post says

The log-likelihood is, as the term suggests, the natural logarithm of the likelihood.

In turn, given a sample and a parametric family of distributions (i.e., a set of distributions indexed by a parameter) that could have generated the sample, the likelihood is a function that associates to each parameter the probability (or probability density) of observing the given sample.

I cannot imagine what "a set of distributions indexed by a parameter" is.

Is it something like a set of different normal distributions?

For example, $X_{\theta_1} \sim {\mathcal {N}}(\mu_1 ,\sigma_1 ^{2})$, $X_{\theta_2} \sim {\mathcal {N}}(\mu_2 ,\sigma_2 ^{2})$ ... where the parameter vector is $\theta = [\mu, \sigma^{2}]$

Does "a set of different normal distributions" imply this kind of families?

Could some give an examples of "a set of distributions indexed by a parameter"?

The term "indexed" is the most confusing part, which reminds me something like a sequence of id {1, 2, ...}

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For example, the set of all functions $f$ such that $$ f(x) = \begin{cases} \lambda e^{- \lambda x} & \quad \text{if } x \geq 0,\\ 0 & \quad \text{otherwise} \end{cases} $$ for some number $\lambda > 0$.

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  • $\begingroup$ Thank you! Is or is not my example $X_{\theta_1} \sim {\mathcal {N}}(\mu_1 ,\sigma_1 ^{2})$, $X_{\theta_2} \sim {\mathcal {N}}(\mu_2 ,\sigma_2 ^{2})$ a parametric family of distributions? $\endgroup$
    – JJJohn
    Oct 12, 2019 at 9:55
  • $\begingroup$ @baojieqh That example is fine. Ctrl+F to "We want to solve" here. $\endgroup$
    – J.G.
    Oct 12, 2019 at 10:31

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