| Date | November 2016 | Marks available | 10 | Reference code | 16N.3sp.hl.TZ0.4 |
| Level | HL only | Paper | Paper 3 Statistics and probability | Time zone | TZ0 |
| Command term | Prove that | Question number | 4 | Adapted from | N/A |
Question
Two independent discrete random variables \(X\) and \(Y\) have probability generating functions \(G(t)\) and \(H(t)\) respectively. Let \(Z = X + Y\) have probability generating function \(J(t)\).
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) \({\text{Var}}(Z) = {\text{Var}}(X) + {\text{Var}}(Y)\).
Markscheme
\(J(t) = G(t)H(t)\) 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]
