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

Q:

From: JJJohn

asked Oct 12, 2019 at 9:23, edited Jun 12, 2020 at 10:38

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

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,

\(\large X_{\theta_1} \sim {\mathcal {N}}(\mu_1 ,\sigma_1 ^{2})\),

\(\large X_{\theta_2} \sim {\mathcal {N}}(\mu_2 ,\sigma_2 ^{2})\),

...

parameter vector is \(\large \theta = [\mu, \sigma^{2}]\)

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 \(\large {1, 2, \cdots}\)

A

From: littleO

answered Oct 12, 2019 at 9:44

For example, the set of all functions \(\large f\), such that

$\large f(x) = \begin{cases} \lambda e^{- \lambda x} & \quad \text{if } x \geq 0,\ 0 & \quad \text{otherwise}

\end{cases} $ ,otherwise for some number \(\large \lambda > 0\)

Reply by JJJohn:

Thank you! Is or is not my example \(\large X_{\theta_1} \sim {\mathcal {N}}(\mu_1 ,\sigma_1 ^{2})\), \(\large X_{\theta_2} \sim {\mathcal {N}}(\mu_2 ,\sigma_2 ^{2})\), a parametric family of distributions? –

That example is fine. Ctrl+F to "We want to solve" here.