Piecewise PDF with multiple regions

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 5 } x & 0 \leqslant x < 2
\frac { 2 } { 15 } ( 5 - x ) & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
2. Find the median value of \(X\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Find \(\mathrm { P } ( 1 \leqslant x \leqslant 3 )\).

7
\includegraphics[max width=\textwidth, alt={}, center]{a93e5413-6ad8-4957-8efd-470cf79792e2-12_428_693_260_724} A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} ( \sqrt { } 2 ) \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi
0 & \text { otherwise } \end{cases}$$ as shown in the diagram.

1. Find \(\mathrm { P } \left( X > \frac { 1 } { 6 } \pi \right)\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4
\includegraphics[max width=\textwidth, alt={}, center]{c7cbd61b-9a62-494a-b595-f624ec5c0bea-2_351_561_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).

4
\includegraphics[max width=\textwidth, alt={}, center]{0784d885-5710-4eb4-8cf8-2582122bf7ed-2_351_554_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\).
For \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant. For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ). For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find

1. the mode,
2. the value of \(k\),
3. the value of \(a\). Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
4. state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).

4 A continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant 3
9 k & 3 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Show that the value of \(k\) is \(\frac { 1 } { 18 }\).
3. 1. Write down the median value of \(X\).
   2. Calculate the value of the lower quartile of \(X\).

7 The time, \(T\) hours, that the supporters of Bracken Football Club have to queue in order to obtain their Cup Final tickets has the following probability density function. $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 5 } & 0 \leqslant t < 3
\frac { 1 } { 45 } t ( 6 - t ) & 3 \leqslant t \leqslant 6
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Write down the value of \(\mathrm { P } ( T = 3 )\).
3. Find the probability that a randomly selected supporter has to queue for at least 3 hours in order to obtain tickets.
4. Show that the median queuing time is 2.5 hours.
5. Calculate P (median \(< T <\) mean).

4 The delay, in hours, of certain flights from Australia may be modelled by the continuous random variable \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 2 } { 15 } t & 0 \leqslant t \leqslant 3
1 - \frac { 1 } { 5 } t & 3 \leqslant t \leqslant 5
0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Calculate:
   1. \(\mathrm { P } ( T \leqslant 2 )\);
   2. \(\mathrm { P } ( 2 < T < 4 )\).
3. Determine \(\mathrm { E } ( T )\).

7 A continuous random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } ( 4 - x ) & 1 \leqslant x \leqslant 3
\frac { 1 } { 6 } & 3 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$

1. Draw the graph of f on the grid on page 6 .
2. Prove that the mean of \(X\) is \(2 \frac { 5 } { 9 }\).
3. Calculate the exact value of:
   1. \(\mathrm { P } ( X > 2.5 )\);
   2. \(\mathrm { P } ( 1.5 < X < 4.5 )\);
   3. \(\mathrm { P } ( X > 2.5\) and \(1.5 < X < 4.5 )\);
   4. \(\mathrm { P } ( X > 2.5 \mid 1.5 < X < 4.5 )\).
      \includegraphics[max width=\textwidth, alt={}, center]{bc21c177-6cd8-4c79-8782-d17f0238ce17-6_1340_1363_317_383}