Write down an expression for $J(t)$ in terms of $G(t)$ and $H(t)$.

By differentiating $J(t)$, prove that

(i)     ${\text{E}}(Z) = {\text{E}}(X) + {\text{E}}(Y)$;

(ii)     ${\text{Var}}(Z) = {\text{Var}}(X) + {\text{Var}}(Y)$.

$J(t) = G(t)H(t)$    A1

[1 mark]

(i)     $J'(t) = G'(t)H(t) + G(t)H'(t)$     M1A1

$J'(1) = G'(1)H(1) + G(1)H'(1)$    M1

$J'(1) = G'(1) + H'(1)$    A1

so $E(Z) = E(X) + E(Y)$     AG

(ii)     $J''(t) = G''(t)H(t) + G'(t)H'(t) + G'(t)H'(t) + G(t)H''(t)$     M1A1

$J''(1) = G''(1)H(1) + 2G'(1)H'(1) + G(1)H''(1)$

$= G''(1) + 2G'(1)H'(1) + H''(1)$    A1

${\text{Var}}(Z) = J''(1) + J'(1) - {\left( {J'(1)} \right)^2}$    M1

$= G''(1) + 2G'(1)H'(1) + H''(1) + G'(1) + H'(1) - {\left( {G'(1) + H'(1)} \right)^2}$    A1

$= G''(1) + G'(1) - {\left( {G'(1)} \right)^2} + H''(1) + H'(1) - {\left( {H'(1)} \right)^2}$    A1

so ${\text{Var}}(Z) = {\text{Var}}(X) + {\text{Var}}(Y)$     AG

Note: If addition is wrongly used instead of multiplication in (a) it is inappropriate to give FT apart from the second M marks in each part, as the working is too simple.

[10 marks]