Covariance formula with CDF (Hoeffding's Covariance Identity)

A complete proof of above lemma can be found on page 241 (Lemma 7.27) of Quantitative Risk Management: Concepts, Techniques and Tools.

Hint: 2\(cov(X_1, X_2) = E[(X_1-\tilde{X_1})(X_2-\tilde{X_2})]\),
where \( (\tilde{X_1}, \tilde{X_2}) \) is an independent copy with the same joint distribution function as  \( (X_1, X_2) \).

Link to MathJax

Mathematical Programming and its modeling langues

Python is widely used in Mathematical Programming as a modeling language.




In commercial products, Gurobi has built its interactive shell in Python.




In open source world, Pyomo from Sandia National Lab use Python to offer an AMPL-like modeling language. Pyomo uses GLPK solver by default, but other solvers, such as GLPK, Gurobi, COIN CBC, can also be selected.




GLPK (GNU Linear Programming Toolkit) supports MathProg, which is also referred as GMPL (GNU Mathematical Programming Language). GLPK provides a command line tool glpsol, which is convenient for users to solve various optimization problems with well designed reports.




PuLP is an LP modeler written in Python. PuLP can generate MPS or LP files, and can call GLPK, COIN CLP.CBC, CPLEX and Gurobi to solve linear problems.


SolverStudio makes it easy to develop models inside Excel using Python. Data entered into the spreadsheet is automatically available to the model. SolverStudio supports PuLP, COOPR/Pyomo, AMPL, GMPL, GAMS, Gurobi, CMPL, SimPy.

OLS in Python

There are a few ways to perform Linear Regression(OLS) in Python.


Here is a short list of them:


1: > pandas.ols(y, x)
2: > pandas.stats.api.ols(y ,x)
3: > scipy.stats.linregress(x, y)
4: > import statsmodels.formula.api as smf
    > results = smf.ols('Close ~ Open + Volume', data = df) # df is a DataFrame