Introduction:
Probabilistic model is the mathematical structure of any phenomenon whose input or output both are uncertain in nature they depend over some particular criteria or region.
Probabilistic experiments:
1. Probability theory is a mathematical framework that allows us to reason about phenomena or experiments whose outcome is uncertain.
2. A probabilistic model is a mathematical model of a probabilistic experiment that satisfies certain mathematical properties (the axioms of probability theory), and which allows us to calculate probabilities and to reason about the likely outcomes of the experiment.
3. A probabilistic model is defined formally by a triple (Ω, F, P), called a probability space, comprised of the following three elements:
a) Ω is the sample space, the set of possible outcomes of the experiment.
b) F is an σ-field, a collection of subsets of Ω.
c) P is a probability measure, a function that assigns a nonnegative probability to every set in the σ-field F.
Our objective is to describe the three elements of a probability space, and explore some of their properties.
Sample space:
1. The sample space is a set Ω comprised of all the possible outcomes of the experiment.
2. Typical elements of Ω are often denoted by ω, and are called elementary outcomes, or simply outcomes.
3. The sample space can be finite, e.g., Ω = {ω1, . . . , ωn}, countable, e.g., Ω = N, or uncountable, e.g., Ω = R or Ω = {0, 1}∞.
4. As a practical matter, the elements of Ω must be mutually exclusive and collectively exhaustive, in the sense that once the experiment is carried out, there is exactly one element of Ω that occurs.
Examples
a) If the experiment consists of a single roll of an ordinary die, the natural sample space is the set Ω = {1, 2, . . . , 6}, consisting of 6 elements. The outcome ω = 2 indicates that the result of the roll was 2.
b) If the experiment consists of five consecutive rolls of an ordinary die, the natural sample space is the set Ω = {1, 2, . . . , 6}5 . The element ω = (3, 1, 1, 2, 5) is an example of a possible outcome.
c) If the experiment consists of an infinite number of consecutive rolls of an ordinary die, the natural sample space is the set Ω = {1, 2, . . . , 6}∞. In this case, an elementary outcome is an infinite sequence, e.g., ω = (3, 1, 1, 5, . . .). Such a sample space would be appropriate if we intend to roll a die indefinitely and we are interested in studying, say, the number of rolls until a 4 is obtained for the first time.
d) If the experiment consists of measuring the velocity of a vehicle with infinite precision, a natural sample space is the set R of real numbers.
