Find quantiles from CDF

2
\includegraphics[max width=\textwidth, alt={}, center]{43b2498f-73e2-4d33-adaf-fc3e460fa36a-2_358_1093_495_520} A random variable \(X\) takes values between 0 and 4 only and has probability density function as shown in the diagram. Calculate the median of \(X\).

5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0
1 - \frac { 1 } { 144 } ( 12 - x ) ^ { 2 } & 0 \leqslant x \leqslant 12
1 & x > 12 \end{cases}$$

1. Find the upper quartile of \(X\).
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).
   The random variable \(Y\) is given by \(Y = \sqrt { X }\).
3. Find the probability density function of \(Y\).

5 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 4 } { 3 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 2 ,
1 & x > 2 . \end{cases}$$

1. Find the median value of \(X\).
2. Find the (cumulative) distribution function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), and hence find the probability density function of \(Y\).
3. Evaluate \(\mathrm { E } \left( 2 - \frac { 2 } { X ^ { 2 } } \right)\).

4. The lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
\frac { 4 } { 9 } \left( x ^ { 2 } + 2 x - 3 \right) & 1 \leqslant x \leqslant 1.5
1 & x > 1.5 \end{array} \right.$$

1. Find the median of \(X\), giving your answer to 3 significant figures.
2. Find, in full, the probability density function of the random variable \(X\).
3. Find \(\mathrm { P } ( X \geqslant 1.2 )\) A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
4. Find the probability that the lantern will still be working after 12 hours.

7 The waiting time, \(X\) minutes, for fans to gain entrance to see an event may be modelled by a continuous random variable having the distribution function defined by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 2 } x & 0 \leqslant x \leqslant 1
\frac { 1 } { 54 } \left( x ^ { 3 } - 12 x ^ { 2 } + 48 x - 10 \right) & 1 \leqslant x \leqslant 4
1 & x > 4 \end{cases}$$

1. 1. Sketch the graph of F.
   2. Explain why the value of \(q _ { 1 }\), the lower quartile of \(X\), is \(\frac { 1 } { 2 }\).
   3. Show that the upper quartile, \(q _ { 3 }\), satisfies \(1.6 < q _ { 3 } < 1.7\).
2. The probability density function of \(X\) is defined by $$\mathrm { f } ( x ) = \begin{cases} \alpha & 0 \leqslant x \leqslant 1
   \beta ( x - 4 ) ^ { 2 } & 1 \leqslant x \leqslant 4
   0 & \text { otherwise } \end{cases}$$
   1. Show that the exact values of \(\alpha\) and \(\beta\) are \(\frac { 1 } { 2 }\) and \(\frac { 1 } { 18 }\) respectively.
   2. Hence calculate \(\mathrm { E } ( X )\).

6 The time, in weeks, that a patient must wait to be given an appointment in Holmsoon Hospital may be modelled by a random variable \(T\) having the cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0
\frac { t ^ { 3 } } { 216 } & 0 \leqslant t \leqslant 6
1 & t > 6 \end{cases}$$

1. Find, to the nearest day, the time within which 90 per cent of patients will have been given an appointment.
2. Find the probability density function of \(T\) for all values of \(t\).
3. Calculate the mean and the variance of \(T\).
4. Calculate the probability that the time that a patient must wait to be given an appointment is more than one standard deviation above the mean.

4. A continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0
\frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right)
1 \end{array} \right.$$ $$\begin{aligned} & x < 4 ,
& 4 \leq x \leq 10 ,
& x > 10 . \end{aligned}$$

1. Find the median value of \(X\).
2. Find the interquartile range for \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
4. Sketch the graph of \(\mathrm { f } ( x )\) and hence write down the mode of \(X\), explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}

7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 ,
\mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 ,
\mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$

1. Show that 0.77 is approximately the median value of \(T\).
2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
3. Find, and sketch, the probability density function of \(T\).
4. Hence find the modal value of \(T\).
5. Comment on the validity of this model.

1. The continuous random variable \(X\) has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 1
1.5 x - 0.25 x ^ { 2 } - 1.25 & 1 \leqslant x \leqslant 3
1 & x > 3 \end{array} \right.$$

1. Find the exact value of the median of \(X\)
2. Find \(\mathrm { P } ( X < 1.6 \mid X > 1.2 )\) The random variable \(Y = \frac { 1 } { X }\)
3. Specify fully the cumulative distribution function of \(Y\)
4. Hence or otherwise find the mode of \(Y\)