In this video I go over a brief introduction on the most commonly used probability density function, the normal distribution. This function is represented by a bell-shaped curve, often referred to as the Bell Curve, and models many natural phenomena well, such as test scores, rainfall, heights, weights, etc. Although I go over the basics of the normal distribution function and its properties in terms of the standard deviation, I leave the actual derivation of the formula to later videos, as it is more advanced. So stay tuned for those videos!
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Applications of Integrals: Probability: Normal Distributions
Many important random phenomena are modeled by a normal distribution (or as described in my earlier videos, as a Bell Curve).
- Examples are:
- test scores on aptitude tests
- heights and weights of individuals from a homogeneous population
- annual rainfall in a given location
This means that the probability density function of the random variable X is a member of the family of functions:
Where:
- μ is the mean or average
- σ is the standard deviation
- σ is the lowercase Greek letter sigma.
The standard deviation is a measure of how spread out the values of X are.
From the bell-shaped graphs of the members of the family in the figure below, we see that for small values of σ the values of X are clustered about the mean, whereas for larger values of σ the values of X are more spread out.
Statisticians have methods for using sets of data to estimate μ and σ.
The factor
is needed to make the f a probability density function.
In fact, it can be verified using the methods of multivariable calculus that:
I will prove this formula and that μ is indeed the average, as well how the normal distribution formula was derived in later videos so stay tuned!