HL Paper 3
Let c be a positive, real constant. Let G be the set \(\{ \left. {x \in \mathbb{R}} \right| - c < x < c\} \) . The binary operation \( * \) is defined on the set G by \(x * y = \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}}\).
Simplify \(\frac{c}{2} * \frac{{3c}}{4}\) .
State the identity element for G under \( * \).
For \(x \in G\) find an expression for \({x^{ - 1}}\) (the inverse of x under \( * \)).
Show that the binary operation \( * \) is commutative on G .
Show that the binary operation \( * \) is associative on G .
(i) If \(x,{\text{ }}y \in G\) explain why \((c - x)(c - y) > 0\) .
(ii) Hence show that \(x + y < c + \frac{{xy}}{c}\) .
Show that G is closed under \( * \).
Explain why \(\{ G, * \} \) is an Abelian group.
A random variable \(X\) has probability density function
\(f(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{1}{2}}&{0 \le x < 1} \\ {\frac{1}{4}}&{1 \le x < 3} \\ 0&{x \ge 3} \end{array}} \right.\)
Sketch the graph of \(y = f(x)\).
Find the cumulative distribution function for \(X\).
Find the interquartile range for \(X\).