Independent solution

How to solve this Expected Value question

Setup

Setup

Represent the three state probabilities and the two decimal price-change vectors.

p=(0.30,0.50,0.20)\mathbf p=(0.30,0.50,0.20)
x=(0.18,0.08,−0.13),y=(0.15,0.07,−0.06)\mathbf x=(0.18,0.08,-0.13),\qquad \mathbf y=(0.15,0.07,-0.06)

Model

Model

Compute the two probability-weighted means.

E[X]=0.30(0.18)+0.50(0.08)+0.20(−0.13)=0.068E[X]=0.30(0.18)+0.50(0.08)+0.20(-0.13)=0.068
E[Y]=0.30(0.15)+0.50(0.07)+0.20(−0.06)=0.068E[Y]=0.30(0.15)+0.50(0.07)+0.20(-0.06)=0.068

Compute

Compute

Weight the squared deviations from the common mean.

Var⁡(X)=∑ipi(xi−0.068)2=0.011676\operatorname{Var}(X)=\sum_i p_i(x_i-0.068)^2=0.011676
Var⁡(Y)=∑ipi(yi−0.068)2=0.005296\operatorname{Var}(Y)=\sum_i p_i(y_i-0.068)^2=0.005296

Answer

Answer

The means are equal, and Stock X has the larger variance.

E[X]=E[Y],Var⁡(X)>Var⁡(Y)(B)\boxed{E[X]=E[Y],\quad \operatorname{Var}(X)>\operatorname{Var}(Y)\quad\text{(B)}}