Let $X_1, X_2, \cdots, X_m$ be random samples from a normal distribution $N(\theta_1, \theta_2)$.
Then,
$$ L(\theta_1,\theta_2) = P(X_1=x_1;X_2=x_2;\cdots;X_n=x_m) = \prod_{i=1}^{m} \dfrac{1}{\sqrt{2\pi\theta_2}}{\text{exp}}{\Big[ -\dfrac{ (x_i-\theta_1)^2 }{2\theta_2} \Big]} \tag{1} $$
To relate to my doubt, I will shrink sample size to $m=2$. That is, we have $X_1,X_2$. Then,
$$ L(\theta_1,\theta_2) = f_{X}(x_1)f_{X}(x_2) = P(X_1=x_1;X_2=x_2) = \prod_{i=1}^{2} \dfrac{1}{\sqrt{2\pi\theta_2}}{\text{exp}}{\Big[ -\dfrac{ (x_i-\theta_1)^2 }{2\theta_2} \Big]} \tag{2} $$
But as per product distributions wiki, if $X_1$ and $X_2$ are two independent continuous random variables, both described by probability density function $f_{X}$, then the probability density function of $Z = X_1X_2$ would be
$$ f_Z(z) = \int\limits_{-\infty}^{\infty}f_{X}(x_1)f_{X}(z/x_1)\dfrac{1}{|x_1|}dx_1 \tag{3} $$
My questions:
1) Does eq.{2} also represent case $Z = X_1X_2$? If so, how do I reduce {3} to {2}?
2) If not, how are {2} and {3} are not related?
3) Is product distribution of $Z=X_1X_2$ same as joint probability distribution $f_{X}(x_1,x_2)$? What are the relation between the two.
Context:
A related question was posted here, from which this doubt arose. Now I am confused, why eq {3} did not interfere in my MLE case.
I took above MLE example from here, page 260.
Absolutely not. There's no reason for you to think that. The product of densities (or product of CDFs) is not the density of product of random variables.
I don't see anywhere in the previous post you linked that suggested that.