matematicas visuales visual math
Normal distribution: Symmetric intervals

It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.

Probability for an Interval = Area under the density curve in that interval

On the basis of a probability density function, you can calculate the probability that the random variable falls within a given range by estimating the area under the curve for that range. With the cumulative probability function you can do the same, but then more accurately

Two points on the x-axis determine the extremes of the interval for which the probability is calculated (approximately)

Different options A1, A2, ..., A6 correspond to different intervals that can be defined with those two points. Taken in pairs these are complementary in the sense that the sum of probabilities is 1.

We can modify the parameters of the normal distribution and see how the probabilities vary.

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


George Marsaglia's article Evaluating the Normal Distribution.


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