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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:
1
f ( x) 
e
2
 1  x  2 
 
 
 2    
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

2
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)