Standard +0.8 This is a multi-part question requiring systematic algebraic manipulation of expectation and variance formulas, optimization using calculus (finding maximum variance), and conceptual understanding that minimum variance occurs when the distribution is degenerate. The optimization in (b)(ii) and the conceptual insight needed for (c) elevate this above routine exercises, though the individual techniques are standard for Further Statistics.
6 The probability distribution of a discrete random variable, $X$, is shown in the table below.
\begin{center}
\begin{tabular}{ | l | c | l | l | }
\hline
$x$ & 0 & 1 & 2 \\
\hline
$\mathrm { P } ( X = x )$ & $1 - a - b$ & $a$ & $b$ \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { E } ( X )$ in terms of $a$ and $b$.
\item \begin{enumerate}[label=(\roman*)]
\item In the case where $\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4$, find an expression for $\operatorname { Var } ( X )$ in terms of $a$.
\item In this case, show that the greatest possible value of $\operatorname { Var } ( X )$ is 0.65 . You must state the associated value of $a$.
\end{enumerate}\item You are now given instead that $\mathrm { E } ( X )$ is not known.
\begin{enumerate}[label=(\roman*)]
\item State the least possible value of $\operatorname { Var } ( X )$.
\item Give all possible pairs of values of $a$ and $b$ which give the least possible value of $\operatorname { Var } ( X )$ stated in part (c)(i).
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Minor 2024 Q6 [9]}}