If the sample space is the ordered set of the n discrete real values x_{k} that can can be taken by the real variable x, a real function p(x_{k}) such that
is said a discrete probability distribution and x is said a random or stochastic variable.
For simplicity's sake, in this section we let
then
Given a discrete probability distribution, two most important descriptors of its global properties are the mean (or expected value) and the variance.
The mean is given by
If we write as ξ_{k} the deviation of the value x_{k} from the mean,
the variance, written as σ^{2} is the mean of the squares of the deviations:
The square root of the variance is said standard deviation of the population:
The standard deviation is important because it gives an evaluation of the spread of the values x_{k} around the mean: the bigger σ the bigger is the dispersion.
Theorem: The variance is the difference between the mean of the squares and the square of the mean.
Proof
When the n values p_{k} are all equal the distribution is said uniform.
From the second of the equalities (3.1) we get
In particular, if x_{k}=k, the mean is
We have applied the equality
known from the theory of arithmetic progressions.
From the equality (3.6), the variance is
We have applied the equality
obtained by induction.
The standard deviation is
Example
In a roll of a fair dice each of the six outcomes k has probability .
The mean value is
The variance is
and the standard deviation
Given the number of trials n and the probability p, the function p_{k}=P_{n,k} in the equality (2.9) represents the probability distribution of the random variable k that can take integer values from 0 to n because this function satisfies the conditions (3.1.1). In fact, since p+q=1 by hypothesis,
Such probability distribution is said binomial distribution or Bernoulli distribution.
From the equality (3.2) the mean value of the binomial random variable k over n trials is
If we expand the right side we have
and finally we get
Moreover we have
The expansion of the right side gives
The expression in brackets can be divided into two sums
s_{2} is the mean value of the binomial random variable k over n-1 trials, so from (3.10),
Then we get
and finally
The following JavaScript application allows you to calculate and to graph a binomial distribution.
The probability p can be expressed either as a decimal or as a fraction.
The values on the ordinate are expressed as a percentage. To view the tables, your browser must allow popups.
If your browser does not allow internal frames, you can directly access the application page.
As p approaches 0 and n approaches infinity, the binomial distribution converges to the Poisson distribution.
In this case, if λ represents the mean value of the distribution, we have
The equality (3.14) may be obtained in the following way:
From (2.9) we have
If n→∞ and p→0
then
If p→0, q→1, then, from the equality (3.12),
The following JS application calculates the probability that an event will occur k times in a Poisson distribution of a fixed mean λ
If your browser does not allow internal frames, you can directly access the application page.
The following JavaScript application allows you to calculate and to graph a Poisson distribution.
The values on the ordinate are expressed as a percentage. To view the tables, your browser must allow popups.
If your browser does not allow internal frames, you can directly access the application page.