Question

Need some help with Stat question

Answer 1

Let S = {1,2,3}, X = I{1}, Y = I {2,3}, and Z = I{1,2}. Let W = X - Y + Z.
a) compute W(1)
b) compute W(2)
c) compute W(3)
d) determine whether or not W >= Z

I don't understand the X = I{1}, Y = I {2,3}, and Z = I{1,2} part, what does it equals to?

Answer 2

@kirbykirby

Answer 3

I believe the way it's written that those are indicator random variables, and \(A_i\) be an event,
\[ I_{ A_i}=\begin{cases} 1 & \text{if } A_i \text{ occurs} \\
0 & \text{otherwise}\end{cases}\]
So:
\[X= I_{ \{ 1\}}=\begin{cases} 1 & \text{if } 1 \text{ occurs} \\
0 & \text{otherwise}\end{cases}\] \[Y= I_{ \{ 2,3\}}=\begin{cases} 1 & \text{if } 2 \text{ or }3 \text{ occurs} \\
0 & \text{otherwise}\end{cases}\] \[ Z=I_{ \{ 1,2\}}=\begin{cases} 1 & \text{if } 1 \text{ or }2\text{ occurs} \\
0 & \text{otherwise}\end{cases}\]
So, \[ \begin{align}W(1)&=X(1)-Y(1)+Z(1) \\ &=1-0+1 \\&=2 \end{align}\]
The others basically are the same idea.

Answer 4

Thank you!
If we want to know whether W >= Z or not, should I calculate W(1), W(2), W(3), Z(1), Z(2),Z(3) then compare the values for each cases? If W >= Z for all cases, then we can prove W>=Z? Am I on the right track?

Answer 5

@kirbykirby

Answer 6

Yes indeed @neoc