Given $M$ histograms:
- 1th-histogram can be described by $\mathcal{N}(x_{1}\mu_{y},x_{1}^2\sigma_{y}^2)$, where $x_{1}$ is a constant randomly drawn from $\mathcal{N}(\mu_{x},\sigma_{x}^2)$.
- 2nd-histogram can be described by $\mathcal{N}(x_{2}\mu_{y},x_{2}^2\sigma_{y}^2)$, where $x_{2}$ is a constant randomly drawn from $\mathcal{N}(\mu_{x},\sigma_{x}^2)$. $$\vdots$$
- $M$-th histogram can be described by $\mathcal{N}(x_{M}\mu_{y},x_{M}^2\sigma_{y}^2)$, where $x_{M}$ is a constant randomly drawn from $\mathcal{N}(\mu_{x},\sigma_{x}^2)$.
Then consider a sum of the $M$ histograms i.e., each bin in the sum of histograms represents the sum of the corresponding bins in multiple individual histograms.
Question 1: What kind of probability density distribution (PDF) can describe the sum of $M$ histograms when $M\rightarrow\infty$? Here we always consider that the mixture distribution is normalized. i.e., the area under the sum of histograms is 1.
Question 2: Is the above question equivalent to finding the PDF below?
$$f_{X}(x)=\sum^{\infty}\frac{1}{X\sigma_{y}\sqrt{2\pi}}\exp\left[-\left(\frac{x-X\mu_{y}}{X\sigma_{y}}\right)^2\right]$$, where $X$ is a random variable $\sim\mathcal{N}(\mu_{x},\sigma_{x}^2)$.