The Normal Distribution

What is the Normal Distribution?

The probability density function of the normal distribution is:

f(x) = (1 / $$ \sigma $$$\sqrt{2\pi}$)e-((x – $\mu$)2/2$ \sigma $2)

The exponent (x – $$\mu$$)2 / $$\sigma$$2 can be reasoned about. The numerator measures how far x is from the mean. By squaring the difference, we always get a positive number, thereby discarding the direction, and only caring for the magnitude. By dividing by the variance (the standard deviation squared), we are tying this difference away from the center, to the specific spread of the distribution. When the equation resolves in a small number, it means x is close to the center of the distribution, and therefore, given the bell shaped curve, has a high probability of appearing. The inverse is also true.

Key properties of normal distributions are:

  1. Symmetry around $$\mu$$
  2. Bell shaped
  3. Area under the curve is 1
  4. The emperical rule states
    1. ~68% change of picking a random x that’s within 1 standard deviation of the mean
    2. ~95% change of picking a random x that’s within 2 standard deviation of the mean
    3. ~99.7% change of picking a random x that’s within 3 standard deviation of the mean

The cumulative distionbution of the standard normal distribution is the CDF of the normal distribution with mean = 0 and standard deviation = 1. N(x) = $\int_\infty^x$ (1 / $\sqrt{2\pi}$)e-(t2/2) dt

The Derivation

The Law of Large Numbers

Application

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