Properties of the normal distribution

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The properties of the normal distribution are a set of characteristics that describe the normal distribution.

In other words, the properties of the normal distribution are the reason why this distribution is so versatile and widely used.

Density function

Properties of the normal distribution

The normal distribution is a theoretical model capable of satisfactorily approximating a value of a random variable to a real value. In other words, the normal distribution fits a random variable to a function that depends on the mean and the standard deviation. That is, the function and the random variable will have the same representation but with slight differences.

Given the following independent random variables that follow a normal distribution:

Normally distributed variables

The normal distribution is well known and is used in most cases because much of the assumptions and statistical theory is based on the normal distribution. Notably, the normal distribution is symmetric, it only depends on two parameters and has a single mode (unimodal).

Characteristics of the normal distribution

  1. Symmetric with respect to its mean. In other words, the mean acts as a mirror in the distribution and makes both tails identical and therefore symmetric.
  2. Mean = Mode = Median. The measures of centralization are the same because the distribution is symmetric.
  3. The distribution changes curvature or has inflection points at the points on the horizontal axis:

Intervals

Intervals

4. According to the standard deviations that are added to the mean, its probability can easily be determined:

  • For this interval we know that it will have a probability of 68%. In other words, the values ​​included in the interval and its extremes have a probability of appearing of 68.2%.

Interval 1
  • For this interval we know that it will have a probability of 95%. In other words, the values ​​within the interval and its extremes have a 95% probability of appearing.

Interval 2
  • For this interval we know that it will have a probability of 99%. In other words, the values ​​in the interval and its extremes have a 99% probability of appearing.

Interval 3

Linear operations

5. Linear operations of addition and subtraction.

The normal distribution allows linear combinations with other normal distributions:

  • Let S be the sum of the independent random variables X and W, this will also follow a normal distribution in which the mean will be the sum of the means and the variance will be the sum of the variances.
Sum of normal distributions
  • Let D be the difference or difference of the independent random variables X and W, this will also follow a normal distribution in which the mean will be the difference or difference from the means and the variance will be the sum of the variances.
Subtraction from normal distributions

You can also add parameters that are real numbers:

  • Let h and r be two real numbers, we can make a linear combination of them and an independent variable that follows a normal distribution:
Normal distribution and real numbers

Example

Calculate the probability of the following intervals knowing that the mean is 14 and the standard deviation is 2:

Example

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