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.

2 questions · Moderate -0.6

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CAIE S2 2023 June Q2
8 marks Moderate -0.8
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.
Edexcel S2 2011 January Q5
13 marks Moderate -0.3
A continuous random variable \(X\) has the probability density function f(\(x\)) shown in Figure 1. \includegraphics{figure_1} Figure 1
  1. Show that f(\(x\)) = \(4 - 8x\) for \(0 \leqslant x \leqslant 0.5\) and specify f(\(x\)) for all real values of \(x\). [4]
  2. Find the cumulative distribution function F(\(x\)). [4]
  3. Find the median of \(X\). [3]
  4. Write down the mode of \(X\). [1]
  5. State, with a reason, the skewness of \(X\). [1]