Common Equations
This notebook will display equations commonly found in literature rendered via MathJax (LaTex). Please feel free to add any of your equations with a description to the repo.
Mean
Population Mean
$$
\begin{equation}
\mu = {\Sigma X \over N}
\end{equation}
$$
\begin{verbatim}
%%
\begin{equation}
\mu = {\Sigma X \over N}
\end{equation}
%%
\end{verbatim}
Sample Mean
$$
\begin{equation}
\bar{X} = {\Sigma X \over n}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
\bar{X} = {\Sigma X \over n}
\end{equation}
$$
\end{verbatim}
Variance
Population Variance
$$
\begin{equation}
\sigma^2 = {\Sigma(X-\mu)^2 \over N}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
\sigma^2 = {\Sigma(X-\mu)^2 \over N}
\end{equation}
$$
\end{verbatim}
Sample Variance
$$
\begin{equation}
s^2 = {\Sigma(X-\bar{X})^2 \over (n-1)}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
s^2 = {\Sigma(X-\bar{X})^2 \over (n-1)}
\end{equation}
$$
\end{verbatim}
Standard Deviation
Population
$$
\begin{equation}
\sigma = \sqrt{\Sigma(X-\mu)^2 \over N}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
\sigma = \sqrt{\Sigma(X-\mu)^2 \over N}
\end{equation}
$$
\end{verbatim}
Sample
$$
\begin{equation}
s = \sqrt{\Sigma(X-\bar{X})^2 \over (n-1)}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
s = \sqrt{\Sigma(X-\bar{X})^2 \over (n-1)}
\end{equation}
$$
\end{verbatim}
Covariance
Sample
$$
\begin{equation}
cov(x,y) = {\Sigma(x-\bar{x})(y-\bar{y}) \over (n-1)}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
cov(x,y) = {\Sigma(x-\bar{x})(y-\bar{y}) \over (n-1)}
\end{equation}
$$
\end{verbatim}
Population
$$
\begin{equation}
cov(x,y) = {\Sigma(x-\mu_{x})(y-\mu_{y}) \over N}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
cov(x,y) = {\Sigma(x-\mu_{x})(y-\mu_{y}) \over N}
\end{equation}
$$
\end{verbatim}
Pearson Correlation
Sample
$$
\begin{equation}
r = {\Sigma(z_{x}z_{y}) \over (n-1)}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
\rho = {\Sigma(z_{x}z_{y}) \over (n-1)}
\end{equation}
$$
\end{verbatim}
Population
$$
\begin{equation}
\rho = {\Sigma(z_{x}z_{y}) \over N}
\end{equation}
$$
\begin{verbatim}
$$
\begin{equation}
\rho = {\Sigma(z_{x}z_{y}) \over N}
\end{equation}
$$
\end{verbatim}
$$
H(X)=-\sum _{i=1}^{n}p(x_{i})\log p(x_{i})
$$
\begin{verbatim}
$$
H(X)=-\sum _{i=1}^{n}p(x_{i})\log p(x_{i})
$$
\end{verbatim}
Softmax
$$
P(y=j|\mathbf {x} )={\frac {e^{\mathbf {x} ^{\mathsf {T}}\mathbf {w} _{j}}}{\sum _{k=1}^{K}e^{\mathbf {x} ^{\mathsf {T}}\mathbf {w} _{k}}}}
$$
\begin{verbatim}
$$
P(y=j|\mathbf {x} )={\frac {e^{\mathbf {x} ^{\mathsf {T}}\mathbf {w} _{j}}}{\sum _{k=1}^{K}e^{\mathbf {x} ^{\mathsf {T}}\mathbf {w} _{k}}}}
$$
\end{verbatim}
Jeffreys Prior
$$
\left({\vec \theta }\right)\propto {\sqrt {\det {\mathcal {I}}\left({\vec \theta }\right)}}.\
$$
\begin{verbatim}
$$
\left({\vec \theta }\right)\propto {\sqrt {\det {\mathcal {I}}\left({\vec \theta }\right)}}.\
$$
\end{verbatim}
z Score
Population
$$
\begin{equation}
z = {X-\mu \over \sigma}
\end{equation}
$$
\begin{verbatim}
\begin{equation}
z = {X-\mu \over \sigma}
\end{equation}
\end{verbatim}
Sample
\begin{equation}
z = {X-\bar{X} \over s}
\end{equation}
\begin{verbatim}
\begin{equation}
z = {X-\bar{X} \over s}
\end{equation}
\end{verbatim}