The number of eggs laid by an insect follows a Poisson distribution with an unknown mean $\lambda$. Once laid, each egg has an unknown chance, $p$, of hatching and the hatching of one egg is independent of the hatching of the others.

An entomologist studies a set of $n$ such insects observing both the number of eggs laid and the number of eggs hatching for each nest.

(a) Give a formal statement of the data and probability model for the data. State whether the model in question is parametric or nonparametric.

(b) Are the parameters of the model identifiable?

Will saying that :

As $\Theta$ is in a finite dimension euclidian space namely with we could conclude that the model is parametric

wrong in this case?

The route we followed in class was that since it easy to observe that the joint of this said distribution with

$y_i=$ number of eggs laid

$x_i=$ number of eggs hatched

is $p(x,y)= {y \choose x}p^x(1-p)^{y-x}\frac{e^{-\lambda}\lambda^y}{y!} $ and that since other model of y are plausible hence this model is parametric.

Is it parametric just because changing y-dist could change the whole model?

If so, how would that relate to the idea of having a finite $\Theta$ space lead to your model being parametric?

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    $\begingroup$ You appear to conflate "finite dimensional" with "finite," but that's incorrect. Note, too, that "plausibility" is unrelated to the question of whether a model is parametric or not. $\endgroup$ – whuber Sep 25 '19 at 15:56
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    $\begingroup$ I personally think that the difference between parametric and non-parametric models is somewhat grey. However, the fact that your model is parametrized by variables $p$ and $\lambda$ that we are learning from our data suggests that this is a parametric model. $\endgroup$ – Scott Sep 25 '19 at 17:49

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