Let A,B,C be three events, and let X=Ia,Y=Ib, and Z=Ic be the associated indicator random variables. We already know that X.Y is the indicator random variable of the event A(intersection)B. In the same spirit, give an algebraic expression, involving X,Y and Z for the indicator random variable of the following events.Express your answers in terms of X,Y and Z.
1. The event Ac∩B∩C.
2. At most two of the events A,B,C occurred.
according to me the answer to part 1 is (1-x)*y*z.
correct me if i am wrong
1. The event Ac∩B∩C.
Answer: (1-x)*y*z --Divine you right
2. At most two of the events A,B,C occurred.
Answer: 1-(X*Y*Z)
Please share answers to other questions as well.
To find the algebraic expressions for the indicator random variables of the given events, we need to understand the relationship between the events and their associated indicator random variables.
1. The event Ac∩B∩C:
The indicator random variable for event A is X, and its complement is X'. The intersection of A' (not A), B, and C can be written as (X') ∩ Y ∩ Z. Thus, the algebraic expression for the indicator random variable of the event Ac∩B∩C is (X') ∩ Y ∩ Z.
2. At most two of the events A, B, and C occurred:
To determine if at most two of the events occurred, we can consider the complementary scenario: when three events occurred or more than two events occurred. If three events occurred, then A, B, and C must all occur, which can be expressed as X ∩ Y ∩ Z. If more than two events occurred, at least two of the indicators X, Y, and Z would be equal to 1. So, the complementary event can be expressed as (X ∩ Y') ∪ (X ∩ Z') ∪ (Y ∩ Z') ∪ (X ∩ Y ∩ Z').
Since we want the indicator random variable for the given event (at most two events occurred), we need the complement of the expression above. Therefore, the algebraic expression for the indicator random variable of the event "at most two of the events A, B, and C occurred" is 1 - [(X ∩ Y') ∪ (X ∩ Z') ∪ (Y ∩ Z') ∪ (X ∩ Y ∩ Z')].