Question

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.

Answer 1

1. (1-X)*(1-Z)

be aware your keyboard is set to US language not US international or other language

Answer 2

2) X*(1-X*Y)*(1-X*Z) + Y*(1-X*Y)*(1-Y*Z) + Z*(1-X*Z)*(1-Y*Z)

Answer 3

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)

Answer 4

2. official answer:

1-(1-(X)*(1-Y)*(1-Z))*(1-(1-X)*(Y)*(1-Z))*(1-(1-X)*(1-Y)*(Z))

Answer 5

and if at most two of the events A, B, C occurred?