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Chapter 2.5
Discrete Random Variable

Knowledge Base > Math> Probability Theory

A random variable is a function which assigns numerical values to all possible outcomes (or sample point) of a random experiment under certain conditions. A random variable is not a variable but rather a function that maps events to numbers. The set of values that a random variable X can assume is called space or domain of X or Domain(X). They are mutually exclusive and exhaustive. A probability space P induces a probability distribution for any random variable X, such that

P(X = xi) = ∑ P(x) {x∈Domain(X)=xi}
(2.5.1)

For example, probability space for rolling a dice of odd value is P(Odd = true) = P(1) + P(3) + P(5) = 1∕2

Discrete probability theory deals with events that occur in countable sample space. The set of all possible outcomes is called the sample space, generally denoted as S. An event A is an subset of S (A ⊆S), and each elementary events is mutually exclusive and exhaustive from one another, that is, the probability of event A is:

∑ P (A ) = f(x) x∈A

For each element x ∈S, or sometimes called atomic event or sample point, there is a probability value that satisfies the following properties

∑ P (x) ∈ [0,1] ∀x ∈ Ω P (x) = 1 x∈Ω
That is, the probability mass function P(x) lies between zero and one for every value of x in the sample space S, and the sum of f(x) over all values x in the sample space S is exactly equal to 1.

By definition, the probability of the entire sample space is 1, and the probability of the null event ∅, is 0.
If two events A and B are mutually exclusive, then the probability of both event occurring is P(A ∩ B), often written as P(A,B), called joint probability.
Next, the probability of at least one event occurring is
P (A∪ B ) = P(A) +P (B)
Conversely, if some elements resides in both A and B, then the result would be
P (A ∪ B) = P(A)+ P (B )- P (A ∩ B)
For example, assume every outcome of rolling a dice is independent from each other, then the event that result an even number is A = 2, 4, 6. Then P(A) = P(2) + P(4) + P(6) = 12.

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