Normal Distributions and The Central Limit Theorem


Here are some useful videos from Khan Academy covering the topic.
Video 1

Video 2

Video 3

The Normal Equation

The normal distribution is defined by the following equation: Normal equation. The value of the random variable Y is: Y = { 1/[ σ * sqrt(2π) ] } * e-(x - μ)2/2σ2
where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.1416, and e is approximately 2.7183.

The random variable X in the normal equation is called the normal random variable. The normal equation is the probability density function for the normal distribution.

A Wikipedia Image explaining the Normal Distribution.

The Normal Curve
The graph of the normal distribution depends on two factors - the mean and the standard deviation. The mean of the distribution determines the location of the center of the graph, and the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distributions look like a symmetric, bell-shaped curve, as shown below. The curve on the left is shorter and wider than the curve on the right, because the curve on the left has a bigger standard deviation.

The Central Limit Theorem

The central limit theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.

Here are some notes and examples using the CLT.

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