Standard +0.3 This is a standard application of linear combinations of normal distributions requiring students to form X+Y < Z (equivalently X+Y-Z < 0), calculate the mean and variance of the combined distribution, then find a probability using normal tables. While it requires understanding independence and variance addition, it's a routine S2 technique with straightforward arithmetic and no conceptual surprises.
2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]
\(\mu = 8\) (or \(-8\)) B1 seen or implied – award at early stage
\(\text{Var}(X + Y - Z) = 2.2^2 + 1.5^2 + 1.8^2\) M1 For standardising (accept sd/var mixes, but variance must be a combination of at least 2 of \(X\), \(Y\), \(Z\))
\((= 9.49)\)
\(\frac{0-8}{\sqrt{9.49}} = -2.597\) M1
\(\Phi(-2.597) = 1 - \Phi(2.597)\) A1
\(= 0.0047\) [5] For area consistent with their working
Question 2:
$(X + Y - Z) \sim N(8, \ldots)$ B1
$\mu = 8$ (or $-8$) B1 seen or implied – award at early stage
$\text{Var}(X + Y - Z) = 2.2^2 + 1.5^2 + 1.8^2$ M1 For standardising (accept sd/var mixes, but variance must be a combination of at least 2 of $X$, $Y$, $Z$)
$(= 9.49)$
$\frac{0-8}{\sqrt{9.49}} = -2.597$ M1
$\Phi(-2.597) = 1 - \Phi(2.597)$ A1
$= 0.0047$ [5] For area consistent with their working
2 Each day Samuel travels from $A$ to $B$ and from $B$ to $C$. He then returns directly from $C$ to $A$. The times, in minutes, for these three journeys have the independent distributions $\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)$ and $\mathrm { N } \left( 30,1.8 ^ { 2 } \right)$, respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from $A$ to $B$ and $B$ to $C$ is less than the time for his return journey from $C$ to $A$. [5]
\hfill \mbox{\textit{CAIE S2 2014 Q2 [5]}}