Probability is the measure of the expectation that a event will occur or a statement is true .
Probability theory is the mathematical branch to analysis the Probability with the help of some basic axioms .
The Probability (P) is defined on the measure space ( Ω , P) as :-
Ω : [Sample Space] is the set of all possible outcome
E : is a subset of Ω called an Event
P : 2Ω → [0,1] s.t the following three axioms [ Kolmogorov axioms] will be held :-
1. P(E) ≥ 0
2. P(Ω) = 1
3. For a countable sequence of disjoint ( mutually exclusive ) events E1 , E2 ...
P(E1∪ E2 ..) = P(E1) + P(E2) + ...
It further depicts that :-
1. P(A∪B) = P(A) + P(B) - P(A∩B) and further inclusion exclusion principal .
2. P(φ) = 0
Conditional Probability :
P(A|B) = Probability that event A occur it is known that B occur = P(A∩B)/P(B) [Defined as that ]
In continuous and general case Ω may be uncountable and some of e ⊂ Ω may be with 0 probability . There we introduce a enumeration of events as instances of random variable [X ∈ A ⊆ R] ( Here X = x is an event )and take 2 new term as cdf (cumulative distribution function ) F[x] = P(X ≥ x) and pdf (probability density function) p(x) = dF[x]/dx .
Obviously in continuous case P[X = x] = 0 for some or all X and in discrete case P(X = x) is the probability of the event X = x.
X⊥Y [X is independent of Y] iff P(X∩Y) = P(X)P(Y)
Probability theory is the mathematical branch to analysis the Probability with the help of some basic axioms .
The Probability (P) is defined on the measure space ( Ω , P) as :-
Ω : [Sample Space] is the set of all possible outcome
E : is a subset of Ω called an Event
P : 2Ω → [0,1] s.t the following three axioms [ Kolmogorov axioms] will be held :-
1. P(E) ≥ 0
2. P(Ω) = 1
3. For a countable sequence of disjoint ( mutually exclusive ) events E1 , E2 ...
P(E1∪ E2 ..) = P(E1) + P(E2) + ...
It further depicts that :-
1. P(A∪B) = P(A) + P(B) - P(A∩B) and further inclusion exclusion principal .
2. P(φ) = 0
Conditional Probability :
P(A|B) = Probability that event A occur it is known that B occur = P(A∩B)/P(B) [Defined as that ]
In continuous and general case Ω may be uncountable and some of e ⊂ Ω may be with 0 probability . There we introduce a enumeration of events as instances of random variable [X ∈ A ⊆ R] ( Here X = x is an event )and take 2 new term as cdf (cumulative distribution function ) F[x] = P(X ≥ x) and pdf (probability density function) p(x) = dF[x]/dx .
Obviously in continuous case P[X = x] = 0 for some or all X and in discrete case P(X = x) is the probability of the event X = x.
X⊥Y [X is independent of Y] iff P(X∩Y) = P(X)P(Y)
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