4.4.1.4 So­lu­tion esda-d

Ques­tion:

Have I got a die for you! By means of a small piece of lead in­te­grated into its light-weight struc­ture, it does away with that old-fash­ioned un­cer­tainty. It comes up six every time! What will be the ex­pec­ta­tion value of your throws? What will be the stan­dard de­vi­a­tion?

An­swer:

The ex­pec­ta­tion value will, of course, be 6, every throw is a 6;

$$0 1 + 0 2 + 0 3 + 0 4 + 0 5 + 1 6 = 6,$$

and the stan­dard de­vi­a­tion will, of course, be zero, no throw will de­vi­ate from the value 6;

$$\begin{eqnarray*}\sigma & = & \big[ 0(1-6)^2+0(2-6)^2+0(3-6)^2+ \ & & \quad 0(4-6)^2+0(5-6)^2+1(6-6)^2 \big]^{1/2} \ & = & 0 \end{eqnarray*}$$