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\includegraphics[max width=\textwidth, alt={}, center]{10cf346f-dee2-4223-8caa-2a49f1eaa99f-10_547_880_260_621} A random variable \(X\) has probability density function f , where the graph of \(y = \mathrm { f } ( x )\) is a semicircle with centre \(( 0,0 )\) and radius \(\sqrt { \frac { 2 } { \pi } }\), entirely above the \(x\)-axis. Elsewhere \(\mathrm { f } ( x ) = 0\) (see diagram).

1. Verify that f can be a probability density function.
   \(A\) and \(B\) are the points where the line \(x = \sqrt { \frac { 1 } { \pi } }\) meets the \(x\)-axis and the semicircle respectively.
2. Show that angle \(A O B\) is \(\frac { 1 } { 4 } \pi\) radians and hence find \(\mathrm { P } \left( X > \sqrt { \frac { 1 } { \pi } } \right)\).