Samples

The samples consist of two random variables and three classes of events. Each sample can be called an experiment and the goal is to repeat many experiments to study the statistical analysis which extract the unknown parameters $\vec{\theta}$. The proposed toy Monte Carlo is a pedagogical exercise and the objective is to investigate statistical bias.

The domain for the random variable $x_1$ is $[-3,+3]$, while the domain for $x_2$ is $[-6,+6]$. The analytic or template PDFs used for the three classes are:

1. $p(x_1) = \frac{1}{\sqrt{2\pi}\sigma_1}e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}}$ and $p(x_2) = \frac{1}{\sqrt{2\pi}\sigma_2}e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}}$, where $\mu_1=0$, $\mu_2 = 2.2$, $\sigma_1=0.8$, $\sigma_2=1$. The correlation coefficient $\rho_{12}(\ensuremath {{\cal{C}}}_1)$ between $x_1$ and $x_2$ is tunable.
2. $p(x_1) = e^{-\frac{(x+3)}{4}}$ and $p(x_2) = e^{-\frac{(x+6)}{4}}$ The correlation coefficient $\rho_{12}(\ensuremath {{\cal{C}}}_2)$ is set to zero.
3. $p(x_1) = \frac{1}{\sqrt{2\pi}\sigma_1}e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}}$ and $p(x_2) = \frac{1}{\sqrt{2\pi}\sigma_2}e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}}$, where $\mu_1 = \mu_2 = 1$ and $\sigma_1=\sigma_2=2$. The correlation coefficient $\rho_{12}(\ensuremath {{\cal{C}}}_3)$ is set to zero.

The number of events per class ($N_i$) in each experiment are Poisson distributed with means $\lambda_1=\lambda_2=\lambda_3=300$. The total number of events is $N_{\rm {tot}}=N_1+N_2+N_3$. The number of samples generated is set to 10,000 for the proposed ensemble test study.