Standard +0.3 This is a straightforward discrete probability question requiring standard techniques: finding k by summing probabilities to 1, computing probabilities of sums of independent random variables by enumeration, and basic probability calculations. Part (a) is routine algebra, part (b) requires recognizing impossible outcomes, part (c) involves systematic enumeration of outcomes (16 cases but mechanical), and part (d) is simple probability lookup. While multi-part and requiring careful bookkeeping, it demands no novel insight and uses only standard A-level methods, making it slightly easier than average.
6. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) & x = 0,1,2 \\ k ( x - 2 ) & x = 3 \\ 0 & \text { otherwise } \end{cases}\)
where \(k\) is a positive constant.
a Show that \(k = 0.25\)
Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
b Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\)
c Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
d Find \(\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)\)
6. The discrete random variable $X$ has probability function\\
$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) & x = 0,1,2 \\ k ( x - 2 ) & x = 3 \\ 0 & \text { otherwise } \end{cases}$\\
where $k$ is a positive constant.\\
a Show that $k = 0.25$
Two independent observations $X _ { 1 }$ and $X _ { 2 }$ are made of $X$.\\
b Show that $\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0$\\
c Find the complete probability function for $X _ { 1 } + X _ { 2 }$.\\
d Find $\mathrm { P } \left( 1.3 \leqslant X _ { 1 } + X _ { 2 } \leqslant 3.2 \right)$\\
\hfill \mbox{\textit{SPS SPS SM Statistics 2021 Q6 [8]}}