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Normal Distribution

The Normal distribution is a very important example of a continuous random variable (the variable can take any real value).

We can use the Normal distribution as a tool for computing probabilities. For example, a normal curve can be used as an approximation of binomial probabilities.

Normal approximation to Binomial distribution
In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.

The probability density function of a normal distribution is shaped like a bell and it is symmetrical about its mean. The total area under the normal curve is 1.

Normal distribution: shaped like a bell | matematicasVisuales

The density is concentrated around the mean and becomes very small by moving from the center to the left or to the right of the distribution (the so called "tails" of the distribution). The further a value is from the center of the distribution, the less probable it is to observe that value.

Two parameters determine a normal distribution: the mean and the standard deviation.

Normal distribution: mean, standard deviation and  variance| matematicasVisuales

If a probability variable follows a normal distribution we can write:

Normal distribution: Random variable normally distributed | matematicasVisuales

The mean of the distribution determines the location of the center of the graph.

By changing the mean the shape of the graph does not change, but the graph is translated to the right or to the left.

Normal distribution: mean of the distribution determines the location of the center of the graph| matematicasVisuales

The density function has two inflection points, located one standard deviation away from the mean.

Normal distribution: inflection points | matematicasVisuales

By increasing the standard deviation, (if the mean is the same, then the center of the graph does not change), the shape of the graph changes, the curve es wide, the dispersion is high (The higher the standard deviation the greater the variable dispersion).

Normal distribution: large standard deviation | matematicasVisuales

If the standard deviation is small, the curve is tall and narrow.

Normal distribution: small standard deviation | matematicasVisuales

The simplest example is called the Standard Normal Distribution. It is a special case, when the mean is equal to 0 and the variance is 1.

Normal distribution: Standard Normal Distribution | matematicasVisuales

The (cumulative) distribution function has an S-shape.

Normal distribution: The (cumulative) distribution function has an S-shape | matematicasVisuales

The standard normal distribution function gives the probability that a standard normal variable assumes a value less than x.

Normal distribution: The standard normal distribution function | matematicasVisuales

Normal distribution: mean| matematicasVisuales

We can modifying the parameters of the normal distribution.

The mean is represented by a triangle that can be seen as an equilibrium point. By dragging it we can modify the mean.

Dragging the point on the curve (which is one of the two inflexion points of the curve) we modify the standard deviation.

We can see the cumulative distribution function and how it change by modifiyng the mean (simple translation) and the standard deviation (reflecting greater or lesser dispersion of the variable).

The red dots control the vertical and horizontal scales of the graphic.

The Normal distribution was studied by Gauss (1809). Gauss derived the normal distribution mathematically as the probability distribution of the error of measurements, which he called the normal law of errors. Before Gauss: de Moivre(1738) and Laplace (1774).


George Marsaglia's article Evaluating the Normal Distribution.
Saul Stahl's article The Evolution of the Normal Distribution (Mathematics Magazine, vol. 79, (2006), pp. 96-113. From Mathematical Association of America).


Normal Distributions: (Cumulative) Distribution Function
The (cumulative) distribution function of a random variable X, evaluated at x, is the probability that X will take a value less than or equal to x. In this page we study the Normal Distribution.
Normal Distributions: Probability of Symmetric Intervals
Calculating probabilities of symmetric intervals around the mean of a normal distribution.
Poisson distribution
Poisson distribution is discrete (like the binomial) because the values that can take the random variable are natural numbers, although in the Poisson distribution all the possible cases are theoretically infinite.
Student's t-distributions
Student's t-distributions were studied by William Gosset(1876-1937) when working with small samples.
Binomial distribution
When modeling a situation where there are n independent trials with a constant probability p of success in each test we use a binomial distribution.