Probability & Statistics Ch.2 Section 1 Question 6

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Let $S=\{1,2,3,4\}$ , $X=\textit{I}_{\{1,2\}}$ , $Y=\textit{I}_{\{2,3\}}$ , $Z=\textit{I}_{\{3,4\}}$ . Let $W=X+Y+Z$ .

a) Compute $W(1)$ b) Compute $W(2)$ c) Compute $W(4)$ d) Determine whether or not $W \geq Z$

Solution

(a) Compute $W(1)$

W(1)=X(1)+Y(1)+Z(1) = 1+0+0 = 1

(b) Compute $W(2)$

W(2)=X(2)+Y(2)+Z(2) = 1+1+0 = 2

(c) Compute $W(4)$

W(4)=X(4)+Y(4)+Z(4) = 0+0+1 = 1

(d) Determine whether or not $W \geq Z$

First, we compute $W(3)$ so we know its value for our entire sample space:

W(3)=X(3)+Y(3)+Z(3) = 0+1+1 = 2

Now we compare W to Z for every element in our sample space:

$W(1) \geq Z(1) \rightarrow 1 \geq 0$ ✓ True
$W(2) \geq Z(2) \rightarrow 2 \geq 0$ ✓ True
$W(3) \geq Z(3) \rightarrow 2 \geq 1$ ✓ True
$W(4) \geq Z(4) \rightarrow 1 \geq 1$ ✓ True

Since our relation holds true for all elements of our sample space, the relation is True.

Alternatively, we could have concluded that the range of W is $[1,2]$ and the range of Z is $[0,1]$ , therefore $W \geq Z$ .