### Formulas

Formula for multivariate gaussian distribution

Formula of univariate gaussian distribution

Notes:

- There is normality constant in both equations
- Σ being a positive definite ensure quadratic bowl is downwards
- σ2 also being positive ensure that parabola is downwards

### On Covariance Matrix

Definition of covariance between two vectors:

When we have more than two variable we present them in matrix form. So covariance matrix will look like

- Formula of multivariate gaussian distribution demands Σ to be singular and symmetric positive semidefinite, which in terms means sigma will be symmetric positive semidefinite.
- For some data above demands might not meet

### Derivations

Following derivations are available at [0]:

- We can prove[0] that when covariance matrix is diagonal (i.e there is variables are independent) multivariate gaussian distribution is simply multiplication of single gaussian distribution of each variable.
- It was derived that shape of isocontours (figure 1) is elliptical and axis length is proportional to individual variance of that variable
- Above is true even when covariance matrix is not diagonal and for dimension n>2 (ellipsoids)

### Linear Transformation Interpretation

This was proved in two steps [0]:

Step-1 : Factorizing covariance matrix

Step-2 : Change of variables, which we apply to density function

### References

[0] http://cs229.stanford.edu/section/gaussians.pdf

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