3. Discrete probability distributions


If the sample space is the ordered set of the n discrete real values xk that can can be taken by the real variable x, a real function p(xk) such that

Eqn000.gif

is said a discrete probability distribution and x is said a random or stochastic variable.

For simplicity's sake, in this section we let

Eqn007.gif

then

Eqn008.gif


Mean and variance

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

Eqn001.gif

If we write as ξk the deviation of the value xk from the mean,

Eqn002.gif

the variance, written as σ2 is the mean of the squares of the deviations:

Eqn003.gif

The square root of the variance is said standard deviation of the population:

Eqn004.gif

The standard deviation is important because it gives an evaluation of the spread of the values xk 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.

Eqn005.gif

Proof

Eqn006.gif


Discrete uniform distribution

When the n values pk are all equal the distribution is said uniform.

From the second of the equalities (3.1) we get

Eqn100.gif

In particular, if xk=k, the mean is

Eqn101.gif

We have applied the equality

Eqn102.gif

known from the theory of arithmetic progressions.

From the equality (3.6), the variance is

Eqn103.gif

We have applied the equality

Eqn104.gif

obtained by induction.

The standard deviation is

Eqn105.gif

 

Example

In a roll of a fair dice each of the six outcomes k has probability Eqn106.gif.

The mean value is

Eqn107.gif

The variance is

Eqn108.gif

and the standard deviation

Eqn109.gif

 

 


Discrete binomial distribution (Bernoulli distribution)

Given the number of trials n and the probability p, the function pk=Pn,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,

Eqn200.gif

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

Eqn201.gif

If we expand the right side we have

Eqn202.gif

and finally we get

Eqn203.gif

Moreover we have

Eqn204.gif

The expansion of the right side gives

Eqn205.gif

The expression in brackets can be divided into two sums

Eqn206.gif

Eqn207.gif

s2 is the mean value of the binomial random variable k over n-1 trials, so from (3.10),

Eqn208.gif

Then we get

Eqn209.gif

and finally

Eqn210.gif

Eqn211.gif

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.

 

 


Poisson distribution

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

Eqn300.gif

The equality (3.14) may be obtained in the following way:

If p→0, q→1, then, from the equality (3.12),

Eqn309.gif

 

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.