Verify geometric PDF from graph

Questions that provide a geometric description or graph (straight line, semicircle, or other shape) and ask to verify it's a valid PDF by calculating the area geometrically or through integration.

1 questions

CAIE S2 2023 June Q2
2

  1. \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-04_324_574_264_813} The graph of the function f is a straight line segment from \(( 0,0 )\) to \(( 2,1 )\).
    Show that \(f\) could be a probability density function.

  2. \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-04_364_592_1466_804} The graph of the function g is a semicircle, centre \(( 0,0 )\), entirely above the \(x\)-axis.
    Given that g is a probability density function, find the radius of the semicircle.

  3. \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-05_369_826_264_689} The time, \(X\) minutes, taken by a large number of students to complete a test has probability density function h , as shown in the diagram.
    1. Without calculation, use the diagram to explain how you can tell that the median time is less than 15 minutes.
      It is now given that $$h ( x ) = \begin{cases} \frac { 40 } { x ^ { 2 } } - \frac { 1 } { 10 } & 10 \leqslant x \leqslant 20
      0 & \text { otherwise. } \end{cases}$$
    2. Find the mean time.