Normal Distribution

This is a naturally occurring distribution in many cases. Also known as the bell curve.

Core Idea

If $X$ follows a normal distribution,

$$X \sim \mathcal{N}(\mu, \sigma^2)$$

• $\mu$ - Expected value
• $\sigma$ - Standard Deviation

PDF

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$$

Expected Value

$$E[X] = \mu$$

Variance

$$V[X] = \sigma^2$$

Standard Normal

This is a normalized version of any normal distribution so that, mean is 0, and std. deviation is 1,

if a distribution $Z$ is a standard normal distribution,

$$Z \sim \mathcal{N}(0, 1)$$ $$Z=\frac{X-\mu}{\frac{\sigma}{\sqrt n}}$$