matematicas visuales visual math

Calculating probabilities in Normal distributions

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

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 gray dots control vertical and horizontal scales of the graphic. By pressing the right button and dragging we can move left and right.


George Marsaglia's article Evaluating the Normal Distribution.


Normal distribution
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.
One, two and three standar deviations
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.
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.
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.