Unizor - Probability - Mass Distribution Function

Unizor - Creative Minds through Art of Mathematics - Math4Teens
Notes to a video lecture on http://www.unizor.com

Probability Mass Function

Here we will consider random variables with discrete distributions of probabilities.

Let's assume that random variable ξ represents the results of the following experiment.
We shoot a target that consists of three concentric circles marked with numbers of points assigned to each circle: 10 for the smallest circle in the center, 5 for the middle circle, 2 for the largest outer circle and 0 for outside the largest circle.
So, our random variable ξ in this experiment has values {0,2,5,10}.

Each participant in the sharpshooting competition has different skills and, therefore, each of them can be represented by a random variable that takes the above values but with different distribution of probabilities for different people.

We can describe the sharpshooting skills of any particular participant by the probability of this particular participant to hit each circle.
For instance, for participant A these probabilities might be:
Prob{ξ=0}=0.32
Prob{ξ=2}=0.28
Prob{ξ=5}=0.23
Prob{ξ=10}=0.17
(the sum of all probabilities must be equal to 1, of course)

The above is a kind of verbal description of the values and probabilities for our random variable.
Mathematicians, however, always prefer to deal with something more resembling a function. So, they have come up with a concept of mass distribution function that serves exactly this purpose.
Our task is to represent the distribution of probabilities of this random variable as an algebraic function and graphically.

Consider a function fξ(x) defined for a discrete random variable ξ for all real arguments x as follows:
f(x) = Prob{ξ=x}

It is important to note that this is a function defined for all real arguments x, regardless of whether ξ can or cannot take this particular value of an argument.
For those values of an argument that random variable ξ can take the function's value is a corresponding probability.
For all other values of an argument the function's value is zero since the probability of ξ to take this value is zero.
This function is called mass distribution function (MDF). It is specific for each discrete random variable and completely describes in an algebraic form the distribution of probabilities of this random variable.

In the example above with target shooting the mass distribution function for participant A looks like this:
f(0)=0.32
f(2)=0.28
f(5)=0.23
f(10)=0.17
and for all other arguments x the function value is zero.

Since mass distribution function is a function in algebraic sense, we can analyze it as any other function. The most important way of dealing with this function is to represent it graphically.

This function is equal to 0 for all negative x,
jumps up to 0.32 at x=0,
then it takes the value of 0 for all x between 0 and 2,
jumps up to 0.28 at x=2,
then it takes the value of 0 for all x between 2 and 5,
jumps up to 0.23 at x=5,
then it takes the value of 0 for all x between 5 and 10,
jumps up to 0.17 at x=10,
then it takes the value of 0 for all x greater than 10.

As a conclusion,
mass distribution function (MDF) is a convenient way to represent algebraically and, especially, graphically the distribution of probabilities of discrete random variables.
Always make sure that your mass distribution function takes values that sum up to 1 (or 100% if probability is expressed as a percent).

WARNING
Sum of two mass distribution functions of two discrete random variables IS NOT a mass distribution function of a random variable that is the sum of two original variables. A very superficial reason for this is that a sum of two mass distribution functions is a not a mass distribution function at all since the result of summation of its values will be 2, not 1. Obviously, there are some deeper probabilistic reasons for this.
Similarly, a product of a constant (not equal to 1) and a mass distribution function IS NOT a mass distribution function for a new random variable that is a product of the original variable and that constant.