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Gaussian Distribution
PDF is given by $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ Where
- $\mu$ is the mean
- $\sigma$ is the standard deviation
Unit Normal distribution
- $\mu = 0$
- $\sigma = 1$
Characteristics
- Continuous
- Real values
- Always positive
- Doesn’t end in both direction
Uses
- Many things in the real world resemble a gaussian
- Not many things are exactly a gaussian because it extends forever and includes negative values
- But we can approximate things to a gaussian
- Another important use is the Central Limit Theorem
- Any distribution no matter its type has a normally distributed sample mean
- Gaussian is preserved under addition/subtraction
- Suppose we have two random variables that are normally distributed. Then the sum of those two is also normally distributed