Standard +0.8 This question requires students to recognize that comparing X₁ and X₂ is equivalent to finding P(X₁ - 3X₂ > 0), then apply linear combinations of normal variables to find the mean and variance of this new variable, and finally use standardization. While the individual techniques are standard S2 content, the conceptual leap to reformulate the comparison and handle the coefficient 3 correctly makes this moderately challenging—above average but not exceptionally difficult.
2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)\). Two independent random values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen.
Find \(\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)\).
2 The random variable $X$ has the distribution $\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)$. Two independent random values of $X$, denoted by $X _ { 1 }$ and $X _ { 2 }$, are chosen.
Find $\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)$.\\
\hfill \mbox{\textit{CAIE S2 2024 Q2 [5]}}