Normal distributions are a family of distributions that depends on two parameters: the mean and the variance. The positive square root of the variance is called the standard deviation.

Normal distributions

The Normal distribution was studied by Gauss. This is a continuous random variable (the variable can take any real value). The density function is shaped like a bell.

The standard normal distribution (with mean 0 and stardard deviation 1) is the simplest case:

One important property of normal distributions is that if we consider intervals centered on the mean and a certain extent proportional to the standard deviation, the probability of these intervals is constant regardless of the mean and standard deviation of the normal distribution considered.

For example, if we consider an interval centered on the mean with amplitude one standard deviation up and down, the probability is about 0.68.

This value is the area below the density probability function (the bell curve) between these two values of the variable:

The method we can use to calculate this area is using the (cumulative) distribution function (the area and the probability is the substraction of two values):

We can see that this value does not change if we change the mean an the standard deviation of the normal distribution:

If the amplitude is two standard deviations up and down, the probability is 0.955.

If the amplitude is three standard deviations up and down, the probability is 0.997.

These probability values (0.68, 0.95 and 0.997) for intervals of one, two, and three sigma around the mean of a normally distributed variable are often used in statistical calculations.

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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.

Calculating probabilities in t Student distributions (Spanish)

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.

Normal approximation to Binomial distribution

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