Normal Distribution Help

What do these properties mean?

• Fully Defined by its Mean and Standard Deviation

  This means that two parameters are necessary to know in order to define the entire distribution - the mean and standard deviation. This means that if you know the mean and standard deviation of a normal distribution, you know the entire distribution.

• Binomial Approximation

  As the number of trials conducted in a binomial experiment increases, the resulting distribution approximates the normal distribution. This is because the variability of small sample sizes is decreased as the number of trials becomes larger.

• Random Variable

  In the case of a continuous distribution, the random variable is defined as having the density of the distribution. So, in this case, since the value that you are interested in observing is the random variable, it has the density of the normal distribution.

Example

When dealing with a continuous distribution, if you see or hear the words "Gaussian" or "normally distributed", you are dealing with a normal distribution. This is because these are simply two different ways to refer to the normal distribution.

For example, imagine that you are collecting data about the IQ of a random sample of American adults. You know from previous data that the IQ of American adults is normally distributed with an average of 100 and a standard deviation of 15. This would be a normal distribution with 𝜇 = 100 (because that is the average IQ of an American adult) and 𝜎 = 15 (because that is the standard deviation of the IQ of American adults). The only thing left would be to determine the IQ that you are interested in observing from the distribution, and that would be x (because it is the value you are finding the probability density of). Note that you will not always observe that IQ from your sample - there is only some probability that you will.

Requirements

• The standard deviation of the normal distribution must be greater than zero (𝜎 > 0).