Question

prove that -slly distributed with mean- O and variance - 1, regardless ofthe particular values ofs and

Answer 1

Since $V_t$ has normal distribution with mean $\sqrt{\rho}s$ and variance $1-{\rho}$,so

$Z={V_t-\sqrt{\rho}s\over\sqrt{1-{\rho}}}$ has normal distribution as well.

Now,

$E(Z)=E\left ( {V_t-\sqrt{\rho}s\over\sqrt{1-{\rho}}} \right ) \\ ~~~~~~~~~~~~={E(V_t-\sqrt{\rho}s)\over\sqrt{1-{\rho}}}~~~~~~~~~~\text{property of expectation}\\ ~~~~~~~~~~~~={E(V_t)-\sqrt{\rho}s\over\sqrt{1-{\rho}}}~~~~~~~~~~\text{property of expectation} \\ ~~~~~~~~~~~~={\sqrt{\rho}s-\sqrt{\rho}s\over\sqrt{1-{\rho}}}~~~~~~~~~~~E(V_t)=\sqrt{\rho}s \\ ~~~~~~~~~~~~=0$

Also

$E(Z^2)=E\left ( {(V_t-\sqrt{\rho}s)^2\over{1-{\rho}}} \right ) \\ ~~~~~~~~~~~~={E(V_t-\sqrt{\rho}s)^2\over {1-{\rho}}}~~~~~~~~~~\text{property of expectation} \\ ~~~~~~~~~~~~={1-\rho \over {1-{\rho}}}~~~~~~~~~~~~~~~~~~~Var(V_t)=1-{\rho} \\ ~~~~~~~~~~~~=1$

Hence Varaince of Z is

$Var(Z)=E(Z^2)-[E(Z)]^2=1$