Normal Distribution

• Description: Bell-shaped curve symmetric around the mean. Arises in many natural processes due to the Central Limit Theorem.
• Parameter(s):     - $\mu$: mean     - ${\sigma }^2$: variance
• Use Case: Heights, weights, financial returns (rough approximation).
• Properties: 68-95-99.7 rule for 1, 2, 3 standard deviations.

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The normal distribution’s formula is:

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

Why is the formula like this?

• $e^x$: The function $e^x$ describes exponential growth, and by making the exponent negative $e^{-x}$ it describes an exponential decay. To make this decay in both directions, you need to make sure that the exponent is always negative, for example by squaring it $e^{-x^2}$ or by taking the absolute value $e^{-|x|}$. This gives us the classic bell curve shape.
• If we add a constant $c$ in front of the exponent $e^{-x{c}^2}$, we can describe more narrow or wider bell curves.
• Since the area under the entire bell curve represents the probability of something happening, it should be 1. If we just represent the bell curve as $e^{-x^2}$, then the area is $\sqrt{\pi }$, so to have an area of $1$ we can just use $\frac{1}{\sqrt{\pi }}{e}^{-x^2}$.
• We can reformulate the entire thing as $\frac{1}{\sqrt{\pi }}{e}^{-\frac{1}{2}(\frac{x}{\sigma }{)}^2}$ in order for $\sigma$ to represent the standard deviation of the curve, but in this way we will have an area of $\sigma \sqrt{2}$, that’s why we also divide by that, obtaining the formula: $$\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}(\frac{x}{\sigma }{)}^2}$$
• We can also subtract the mean $\mu$ from the $x$ in order to be able to center the bell curve around any mean, and not always in the $0$.

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statistics resources:

• But what is the Central Limit Theorem? - YouTube (from min 16:00)