Theme 8. Major probability distributions 1. Discrete random variables: binomial distribution. 2. Continuous random variables: normal distribution. 3. Continuous random variables: t distribution. 4. Continuous random variables: Chi Square distribution. 5. Continuous random variables: F distribution.

8.1 Binomial distribution (discrete random variables) It is used when:   1. We have a number n of "experiments" (observations), all independent of each other. 2. In each of these "experiments”, there is a binary outcome (success [p] vs. failure [1-p]) 3. The probability of "success" [p] is the same in every "experiment" Expected value= n * p With high values of n, this distribution approaches the normal distribution.

8.2 Normal distribution (or Gaussian) It is the best known distribution. Its density function is: Where alpha can be any real number, and beta can be any positive real number; the first functions as mean and the second variance.

normal distribution (2) It is symmetric and unimodal Like any other continuous distribution, the area under the curve is 1 (remember that the curve is asymptotic to the X axis standardized normal distribution It is one that has mean 0 and variance 1. It can be expressed as N (0,1)

8.3 Student t Distribution It's symmetrical and unimodal, with mean 0 It's a family of curves, depending on the so-called "degrees of freedom". That is, there is a Student's t distribution with 1 df, Student's t-distribution with 2 df, etc. -As the degrees of freedom increase, the t distribution tends more and more to a standardized normal distribution. (It is used to contrast the means of two groups, etc.)

8.4 Distribution chi-square It never adopts negative values It's positive asymmetrical It's actually a family of curves, depending on the so-called "degrees of freedom". That is, there is a chi-square distribution with 1 df, chi-square distribution with 2 df, etc. (Note: The degrees of freedom are always positive numbers.) -As the degrees of freedom increase, the distribution becomes more and more symmetrical. When to use it: In tests of goodness of fit (when comparing the predicted scores with observed scores), for instance.

8.5 Fisher’s F distribution (in some books "Snedecor’s F") It can not adopt negative values It's positive asymmetrical It's actually a family of curves, depending on the so-called "degrees of freedom" of the numerator and denominator. That is, there is a Fisher F con1 gl in the numerator and 10 df in the denominator, etc. - It can be shown that the distribution F corresponds to a ratio of two chi-square, hence we speak in the case of F degrees of freedom in the numerator and denominator. (Analysis of Variance -ANOVA- among others)