Let A,B,C be three events and let X=IA, Y=IB, Z=IC be the associated indicator random variables. We already know that X⋅Y is the indicator random variable of the event A∩B. In the same spirit, give an algebraic expression, involving X, Y, Z for the indicator random variable of the following events.
The event Ac∩Cc
Exactly one of the events A, B, C occurred.
1. (1-X)*(1-Z)
be aware your keyboard is set to US language not US international or other language
2) X*(1-X*Y)*(1-X*Z) + Y*(1-X*Y)*(1-Y*Z) + Z*(1-X*Z)*(1-Y*Z)
2) Exactly one of the events A, B , C occurs:
[A ∩ (Bc ∩ Cc)] ∪ [B ∩ (Cc ∩ Ac)] ∪ [C ∩ (Ac ∩ Bc)]
Using 1) Indicator for the first formula: X*(1-Y)*(1-Z)
Now we have to use De Morgan Law and finally we have:
1-(1-X*(1-Y)*(1-Z))*(1-(1-X)*Y*(1-Z))*(1-(1-X)*(1-Y)*Z)
2. official answer:
1-(1-(X)*(1-Y)*(1-Z))*(1-(1-X)*(Y)*(1-Z))*(1-(1-X)*(1-Y)*(Z))
and if at most two of the events A, B, C occurred?
To find the algebraic expression for the indicator random variable of the event Ac∩Cc, we can analyze the event and break it down into its components.
1. Event Ac∩Cc:
This event represents the intersection of two events: Ac and Cc, where Ac denotes the complement of event A and Cc denotes the complement of event C.
To find the indicator random variable for Ac∩Cc, we can use the following algebraic expression:
Z(1 - X)
Explanation:
- Z represents the indicator random variable for event C.
- X represents the indicator random variable for event A.
- 1 - X represents the complement of event A, denoted as Ac.
- Z(1 - X) represents the intersection of events Ac and Cc.
For the second part:
2. Exactly one of the events A, B, C occurred:
This event represents the condition where only one of the events A, B, C occurred and the remaining two did not occur.
To find the algebraic expression for the indicator random variable of exactly one of the events A, B, C occurred, we can use the following expression:
(X⋅(1-Y)⋅(1-Z)) + ((1-X)⋅Y⋅(1-Z)) + ((1-X)⋅(1-Y)⋅Z)
Explanation:
- X represents the indicator random variable for event A.
- Y represents the indicator random variable for event B.
- Z represents the indicator random variable for event C.
In this expression, we consider all possible combinations where only one event occurs and the other two do not. The first term X⋅(1-Y)⋅(1-Z) represents the case where event A occurs and events B and C do not. Similarly, the second term (1-X)⋅Y⋅(1-Z) represents the case where event B occurs and events A and C do not. The third term (1-X)⋅(1-Y)⋅Z represents the case where event C occurs and events A and B do not.
These algebraic expressions provide the indicator random variables for the given events in terms of X, Y, and Z.