Tuesday, October 7, 2014

Unizor - Probability - Discrete Distribution

In the previous lectures we introduced a concept of a random experiment and, as its mathematical model, we described sample space Ω (a set) that contained finite number N of elementary events e1, e2,...,eN (elements of this set) modeling the results (outcomes) of our random experiment.

With each such elementary event eK (where K∈[1,N]) we associated certain real number pK, its probability
0 ≤ pK ≤ 1 (K∈[1,N])
ΣpK = 1 where K∈[1,N]

This probability has all the characteristics of a measure (like weight, length or area) - its non-negative and additive - with the only restriction that the sum of all probability measures of all elementary events totals to 1 since it reflects the ratio of occurrence of any outcome to a total number of experiments.

The set of probabilities p1, p2,...,pN is called a probability distribution of our random experiment with finite number of outcomes.

If there is a random variable ξ defined on the set of elementary events that takes the value xK for an event eK (K∈[1,N]), then these same probabilities define the probability distribution of the random variable ξ:
P(ξ=x1) = p1
P(ξ=x2) = p2
P(ξ=xN) = pN

As you see, in this description we address random experiments with finite number of outcomes. The probability distributions associated with these random experiments, their elementary events and random variables defined on these events are called discreet since the different values of probabilities as well as the different values of random variables defined on the elementary events are separated from each other. They can be represented as individual points on a numeric line.

In a more complicated case there might be infinite but countable number of elementary events and values of a random variable defined on them.
For example, an experiment might be to choose any natural number N (this is the elementary event and, at the same time, the value of a random variable ξ defined on it) and assign a value of 1/(2^N) to a probability associated with this elementary event:
There are infinite but countable number of different elementary events, all probabilities are in the range from 0 to 1 and their sum (which is a sum of an infinite geometric progression 1/2 + 1/4 + 1/8 + ...) equals to 1, as can be easily shown.
The distribution of probabilities in this and analogous cases is also considered discrete since there is always a non-zero distance between different measures of probabilities and, if a random variable is defined on these elementary events, the values of such random variable will also be discrete, that is separated from each other.

Incidentally, for the example above would be interesting to calculate the expected value of the random variable ξ, that is to find a sum
Σ[K/(2^K)] where K∈[1,∞).
It's a good problem on geometric progression and we recommend to try to solve it by yourselves. The answer should be 2, by the way, but you need a little trick to come up with it.

It's very useful to represent the distribution of probabilities in a graphic form using a concept of an area as a substitute for a probability measure. On an X-axis in this representation we will use the points 1, 2, 3,... as the corresponding representation of elementary events e1, e2, e3,... This can be used for both finite and infinite countable number of elementary events.
Next on each segment from K−1 to K we build a rectangle of the height equal to the probability of the elementary event eK. The resulting figure - a combination of all such rectangles - is a good representation of the distribution of probabilities among elementary events. It's total area would always be 1 and elementary events with larger probability measure would be represented by higher rectangles.

For example, the distribution of probabilities for an ideal dice would be a set of 6 rectangles of equal height of 1/6.

In an example above, when the probability of choosing the number N equals to 1/(2^N), the picture would be different. The rectangle built on a segment from 1 to 2 will have a height 1/2, from 2 to 3 - 1/4, from 3 to 4 - 1/8 etc. Every next rectangle would have a height equal to a half of a previous one, sloping down to 0 as we move to infinity.

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