Univariate Density Estimation
Parametric Desity Estimation
Draw from a normal distribution
Given \({{X}_{1}},{{X}_{2}},\ldots ,{{X}_{n}}\) \(i.i.d\) draw from a normal distribution with mean of \(\mu\) and variance of \({{\sigma }^{2}}\) the joint \(PDF\) can be expressed as:
\[f\left( {{X}_{1}},{{X}_{2}},\ldots {{X}_{3}} \right)=\prod\limits_{i=1}^{n}{\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{i}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}}\]
\[f\left( {{X}_{1}},{{X}_{2}},\ldots {{X}_{3}} \right)=\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{1}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\times \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{2}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\times \cdots \times \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}{{e}^{-\frac{{{\left( {{X}_{n}}-\mu \right)}^{2}}}{2{{\sigma }^{2}}}}}\]
The term \(\frac{1}{\sqrt{2\pi {{\sigma }^{2}}}}\) is a constant multiplying this term for \(n\) times gives \({{\left( \frac{1}{\sqrt{2\pi {{\sigma }^{2}}}} \right)}^{n}}=\frac{1}{{{\left( 2\pi \sigma \right)}^{\frac{n}{2}}}}\).