Find median or percentiles

5 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 4 }\).
2. Find \(\operatorname { Var } ( X )\).
3. Given that the median of \(X\) is \(m\), find \(\mathrm { P } ( m < X < 3 )\).

5 Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is \(300 + X\), where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4000 } \left( 100 - x ^ { 2 } \right) & - 10 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the probability that a randomly chosen bottle of Lanta contains more than 305 ml of juice.
2. Given that \(25 \%\) of bottles of Lanta contain more than \(( 300 + p ) \mathrm { ml }\) of juice, show that $$p ^ { 3 } - 300 p + 1000 = 0 .$$
3. Given that \(p = 3.47\), and that \(50 \%\) of bottles of Lanta contain between ( \(300 - q\) ) and ( \(300 + q\) ) ml of juice, find \(q\). Justify your answer.

5 \includegraphics[max width=\textwidth, alt={}, center]{c06524f0-a981-48a6-9af0-c4a3474396b3-06_394_723_258_705} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and \(a\) only. It is given that \(\mathrm { P } ( X < 1 ) = 0.25\).

1. Find, in any order,
   1. \(\mathrm { P } ( X < 2 )\),
   2. the value of \(a\),
   3. \(\mathrm { f } ( x )\).
   4. Find the median of \(X\).

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X < 1.2 )\).
2. Find \(\mathrm { E } ( X )\).
   The median of \(X\) is \(m\).
3. Show that \(m ^ { 3 } - 27 m + 27 = 0\).

6 The time in minutes taken by people to read a certain booklet is modelled by the random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2 \sqrt { } t } & 4 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the time within which \(90 \%\) of people finish reading the booklet.
2. Find \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).

5 A continuous random variable, \(X\), has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
   ................................................................................................................................. .
2. Find the median of \(X\).

1 \includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leqslant x < 2 \\ \frac { 1 } { 12 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the upper quartile of \(X\).
2. Find the value of \(a\) for which \(\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )\).

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\). Show that \(Y\) has probability density function given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16 \\ 0 & \text { otherwise } \end{cases}$$ Find

1. the median value of \(Y\),
2. the expected value of \(Y\).

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 114 } ( 4 x + 7 ) & 0 \leq x \leq 6 \\ 0 & \text { otherwise } \end{cases}$$ 6

1. Show that the median of \(X\) is 3.87, correct to three significant figures.
   [0pt] [3 marks]
   6
2. Find the exact value of \(\mathrm { P } ( X > 2 )\)

   	6 The continuous random variable \(Y\) has probability density function \(g ( y ) = \begin{cases} \frac { 1 } { 2 } y ^ { 2 } - \frac { 1 } { 6 } y ^ { 3 } 1 \leq y \leq 3 0 \text { otherwise } \end{cases}\) " 6 (c) (i) Show that \(\operatorname { Var } \left( \frac { 1 } { Y } \right) = \frac { 2 } { 81 }\)	\multirow[b]{2}{*}{ [4 marks] [4 marks] }

6. A random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & 0 \leqslant x < 1 \\ \frac { x ^ { 3 } } { 5 } & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to find \(\mathrm { E } ( X )\)
2. Use algebraic integration to find \(\operatorname { Var } ( X )\)
3. Define the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
4. Find the median of \(X\), giving your answer to 3 significant figures.
5. Comment on the skewness of the distribution, justifying your answer.

5. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k x ( x - 2 ) , & 2 \leq x \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 3 } { 4 }\). Find
2. \(\mathrm { E } ( X )\),
3. the cumulative distribution function \(\mathrm { F } ( x )\).
4. Show that the median value of \(X\) lies between 2.70 and 2.75.

6 The continuous random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} \frac { 3 } { 8 } x ^ { 2 } & 0 \leqslant x \leqslant \frac { 1 } { 2 } \\ \frac { 3 } { 32 } & \frac { 1 } { 2 } \leqslant x \leqslant 11 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Show that:
   1. \(\quad \mathrm { P } \left( X \geqslant 8 \frac { 1 } { 3 } \right) = \frac { 1 } { 4 }\);
   2. \(\quad \mathrm { P } ( X \geqslant 3 ) = \frac { 3 } { 4 }\).
3. Hence write down the exact value of:
   1. the interquartile range of \(X\);
   2. the median, \(m\), of \(X\).
4. Find the exact value of \(\mathrm { P } ( X < m \mid X \geqslant 3 )\).

6 The random variable \(X\) has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 40 } ( x + 7 ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Find the exact value of \(\mathrm { E } ( X )\).
3. Prove that the distribution function F , for \(1 \leqslant x \leqslant 5\), is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 80 } ( x + 15 ) ( x - 1 )$$
4. Hence, or otherwise:
   1. find \(\mathrm { P } ( 2.5 \leqslant X \leqslant 4.5 )\);
   2. show that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 2 } + 14 m - 55 = 0\).
5. Calculate the value of the median of \(X\), giving your answer to three decimal places.

7. The time, in hours, taken to run the London marathon is modelled by a continuous random variable \(T\) with the probability density function $$f ( t ) = \begin{cases} c ( t - 2 ) & 2 \leq t < 4 \\ \frac { 2 c ( 7 - t ) } { 3 } & 4 \leq t \leq 7 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the function \(\mathrm { f } ( t )\), and show that \(c = \frac { 1 } { 5 }\).
2. Calculate the median value of \(T\).
3. Make two critical comments about the model.

12 The continuous random variable \(X\) has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ k \left( a x - 0.5 x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 1 & x > a \end{cases}$$ where \(a\) and \(k\) are positive constants.

1. Determine the median of \(X\) in terms of \(a\).
2. Given that \(a = 10\), determine the probability that \(X\) is within one standard deviation of its mean.

4 An archer fires arrows at a circular target of radius 50 cm . The distance in cm that an arrow lands from the centre of the target is modelled by the random variable \(X\), with probability density function given by \(f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 50 , \\ 0 & \text { otherwise, } \end{cases}\) where \(a\) is a constant.

1. Determine the value of \(a\).
2. Determine the probability that an arrow will land within 5 cm of the centre of the target.
3. Determine the median distance from the centre of the target that an arrow will land.

2. Emlyn aims to produce podcast episodes that are a standard length of time, which he calls the 'target time'. The time, \(X\) minutes, above or below the target time, which he calls the 'allowed time', can be modelled by the following cumulative distribution function. $$F ( x ) = \begin{cases} 0 & x < - 2 \\ \frac { x + 2 } { 5 } & - 2 \leqslant x < 1 \\ \frac { x ^ { 2 } - x + 3 } { 5 } & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$

1. Calculate the upper quartile for the 'allowed time'.
2. Find \(f ( x )\), the probability density function, for all values of \(x\).
3. 1. Calculate the mean 'allowed time'.
   2. Interpret your answer in context.

1. The continuous random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 18 } ( 11 - 2 x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( \mathrm { X } < 3 )\)
2. State, giving a reason, whether the upper quartile of \(X\) is greater than 3, less than 3 or equal to 3 Given that \(\mathrm { E } ( \mathrm { X } ) = \frac { 9 } { 4 }\)
3. use algebraic integration to find \(\operatorname { Var } ( \mathrm { X } )\) The cumulative distribution function of \(X\) is given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 18 } \left( 11 x - x ^ { 2 } + c \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
4. Show that \(\mathrm { c } = - 10\)
5. Find the median of \(X\), giving your answer to 3 significant figures. \section*{Q uestion 2 continued}

5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} x ^ { 3 } & 0 < x \leq 1 \\ \frac { 9 } { 1696 } x ^ { 3 } \left( x ^ { 2 } + 1 \right) & 1 < x \leq 3 \\ 0 & \text { otherwise } \end{cases}$$ 5

1. Find \(\mathrm { P } ( X < 1.8 )\), giving your answer to three decimal places.
   [0pt] [3 marks]
   5
2. Find the lower quartile of \(X\)

   5 (d)

   5 Show that \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = \frac { 133 } { 212 }\)

8 The continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) where $$\mathrm { F } ( x ) = \begin{cases} 0 & x = 0 \\ \mathrm { e } ^ { k x } - 1 & 0 \leq x \leq 5 \\ 1 & x > 5 \end{cases}$$ 8

1. Show that \(k = \frac { 1 } { 5 } \ln 2\) [0pt] [2 marks]
   8
2. Show that the median of \(X\) is \(a \frac { \ln b } { \ln 2 } - c\), where \(a , b\) and \(c\) are integers to be found.
   

   8	Show that the mean of \(X\) is \(p - \frac { q } { \ln 2 }\), where \(p\) and \(q\) are integers to be found.

6 The object distance, \(U \mathrm {~cm}\), and the image distance, \(V \mathrm {~cm}\), for a convex lens of focal length 40 cm are related by the lens law $$\frac { 1 } { U } + \frac { 1 } { V } = \frac { 1 } { 40 } .$$ The random variable \(U\) is uniformly distributed over the interval \(80 \leqslant u \leqslant 120\).

1. Show that the probability density function of \(V\) is given by $$f ( v ) = \begin{cases} \frac { 40 } { ( v - 40 ) ^ { 2 } } & 60 \leqslant v \leqslant 80 \\ 0 & \text { otherwise } \end{cases}$$
2. Find
   1. the median value of \(V\),
   2. the expected value of \(V\).

The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{2t}x^2 & 1 \leq x \leq 4, \\ 0 & \text{otherwise}. \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^2\). Show that \(Y\) has probability density function given by $$g(y) = \begin{cases} \left(\frac{1}{2t}\right)y^{\frac{1}{2}} & 1 \leq y \leq 16, \\ 0 & \text{otherwise}. \end{cases}$$ [5] Find

1. the median value of \(Y\), [2]
2. the expected value of \(Y\). [2]

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

1. Find the median of \(X\), giving your answer to 3 significant figures. [3]
2. Find, in full, the probability density function of the random variable \(X\). [3]
3. Find P(\(X \geqslant 1.2\)) [2]

A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.

1. Find the probability that the lantern will still be working after 12 hours. [2]