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∩B. In the same spirit, give an algebraic expression, involving X,Yand Z, for the indicator random variable of the following events.
Note: Express your answers in terms of X,Y, and Z (the answer box is case sensitive) using standard notation.
1. The event A^c∩B∩C.
2. At most two of the events A,B,C occurred.
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To express the indicator random variable of the given events, we need to determine how the events A, B, and C are related and construct the algebraic expression accordingly.
Let's break down each event and determine the expression:
1. The event A^c∩B∩C:
- The complement of A, denoted as A^c, is the event where A does not occur.
- The intersection of A^c, B, and C represents the event where A does not occur, but both B and C occur simultaneously.
- Therefore, the expression for this event's indicator random variable is (1 - X) * Y * Z.
- The factor (1 - X) represents the complement of A, indicating that A did not occur.
2. At most two of the events A, B, C occurred:
- We want to express the event where either none, one, or two of the events A, B, and C occurred. The event where all three occur is not included.
- The complement of this event corresponds to the event where either all three events occurred or only one of the events occurred.
- Therefore, the expression for this event's indicator random variable is 1 - (X * Y * Z + X * Y * (1 - Z) + X * (1 - Y) * Z + (1 - X) * Y * (1 - Z)).
- The terms X * Y * Z, X * Y * (1 - Z), X * (1 - Y) * Z, and (1 - X) * Y * (1 - Z) represent the events where all three, two, one, and none of the events occurred, respectively.
- Subtracting this sum from 1 gives us the indicator for the event at most two events occurred.
Remember that these expressions are algebraic representations and should be interpreted as such.
1. The indicator random variable for the event A^c∩B∩C: (1 - X) * Y * Z
2. The indicator random variable for at most two of the events A, B, C occurred: 1 - (X * Y * Z + X * Y * (1 - Z) + X * (1 - Y) * Z + (1 - X) * Y * (1 - Z))