Student's tdistributions
Student's tdistributions were discovered by William S. Gosset (18761937) 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 tdistribution 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 tdistribution has n1 degrees of freedom. There is a different tdistribution for each sample size. It is a class or family of continuous probability distributions. The t density curves are symmetric and bellshaped like the standard normal distribution. Its mean is 0 and its variance is bigger than 1 (it has heavier tails. The tails of the tdistributions 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 tdensity to the normal density. With n larger than 30, the difference between the normal and the tdistribution is really not very important. In the image we can see several examples of cumulative distribution functions: In Probabilities in Student's tdistributions you can see a more precise comparation between Student's tdistributions and the standard normal. In the applet we can see several examples of Student's tdistributions and the standard normal distributions. We can see that when the degrees of freedom is 25 the tdistribution is very similar to the standar normal. The gray points control vertical and horizontal scales. Pressing the right button and dragging you can move left and right. LINKS
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 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.
It may be interesting to familiarize ourselves with the probabilities correspondig to different intervals in normal distributions.
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.
In some cases, a Binomial distribution can be approximated by a Normal distribution with the same mean and variance.
