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Student's t-distributions

Student's t-distributions were discovered by William S. Gosset (1876-1937) in 1908 when he was working for the Guinness brewing company in Dublin (Ireland). He could not publish his discoveries using his own name because Guinness prohibited its employees from publishing any papers with confidential information. He then wrote under the pen name "Student". Gosset had a good relationship with Karl Pearson, who had been his teacher. He needed a distribution that he could use when the sample size was small and the variance was unknown and has to be estimated from the data. The t-distribution is often used to account for the extra uncertainty that results from this estimation. Fisher appreciated the importance of Gosset's work with small samples.

If the sample size is n then the t-distribution has n-1 degrees of freedom. There is a different t-distribution for each sample size. It is a class or family of continuous probability distributions. The t density curves are symmetric and bell-shaped like the standard normal distribution. Its mean is 0 and its variance is bigger than 1 (it has heavier tails. The tails of the t-distributions decrease more slowly than the tails of the normal distribution). The larger the degrees of freedom, the closer to 1 is the variance and more similar is the t-density to the normal density.

t-Student distribution: several density functions (different degrees of freedom) | matematicasVisuales

With n larger than 30, the difference between the normal and the t-distribution is really not very important. In the image we can see several examples of cumulative distribution functions:

t-Student distribution: cumulative distribution functions. When degrees of freedom is greater than 30 the Student's t-distribution is similar to the standard normal | matematicasVisuales

In Probabilities in Student's t-distributions you can see a more precise comparation between Student's t-distributions and the standard normal.

In the applet we can see several examples of Student's t-distributions and the standard normal distributions.

We can see that when the degrees of freedom is 25 the t-distribution is very similar to the standar normal.

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