Moderate -0.8 This is a straightforward application of standard results for linear transformations and sums of independent normal random variables. Students need only recall that for X ~ N(μ, σ²): aX ~ N(aμ, a²σ²) and X₁ + X₂ ~ N(μ₁ + μ₂, σ₁² + σ²₂). No problem-solving or conceptual insight required—pure recall and arithmetic.
2 The random variable \(X\) has the distribution \(\mathrm { N } ( 3,1.2 )\). The random variable \(A\) is defined by \(A = 2 X\). The random variable \(B\) is defined by \(B = X _ { 1 } + X _ { 2 }\), where \(X _ { 1 }\) and \(X _ { 2 }\) are independent random values of \(X\). Describe fully the distribution of \(A\) and the distribution of \(B\).
Distribution of \(A\) : \(\_\_\_\_\)
Distribution of \(B\) : \(\_\_\_\_\)
B1 for \(N(6, ...)\) for either A or B. B1 for \(4.8\) (or \(2.19^2\)) (or SD\(=2.19\))
\(B: N(6, 2.4)\)
B1
B1 for \(2.4\) (or \(1.55^2\)) (or SD\(=1.55\)) (SR 3/3 but error seen withhold B1 so 2/3 scored)
Total: 3
**Question 2:**
$A: N(6, 4.8)$ | B1 B1 | B1 for $N(6, ...)$ for either A or B. B1 for $4.8$ (or $2.19^2$) (or SD$=2.19$)
$B: N(6, 2.4)$ | B1 | B1 for $2.4$ (or $1.55^2$) (or SD$=1.55$) (SR 3/3 but error seen withhold B1 so 2/3 scored)
**Total: 3**
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2 The random variable $X$ has the distribution $\mathrm { N } ( 3,1.2 )$. The random variable $A$ is defined by $A = 2 X$. The random variable $B$ is defined by $B = X _ { 1 } + X _ { 2 }$, where $X _ { 1 }$ and $X _ { 2 }$ are independent random values of $X$. Describe fully the distribution of $A$ and the distribution of $B$.
Distribution of $A$ : $\_\_\_\_$\\
Distribution of $B$ : $\_\_\_\_$\\
\hfill \mbox{\textit{CAIE S2 2018 Q2 [3]}}