Single-piece PDF with k

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } x & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find \(a\).
2. Show that \(\mathrm { E } ( X ) = \frac { 4 } { 3 }\).
   The median of \(X\) is denoted by \(m\).
3. Find \(\mathrm { P } ( \mathrm { E } ( X ) < X < m )\).

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

1. Show that \(k = \frac { 2 } { 9 }\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
3. Find \(\operatorname { Var } ( X )\).

4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 2 } { 3 }\).
2. Find the median of \(X\).

5 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 3 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that the value of \(a\) is \(\frac { 1 } { 2 }\).
2. Find \(\mathrm { P } ( X > 1.8 )\).
3. Find \(\mathrm { E } ( X )\).

7 The time in hours taken for clothes to dry can be modelled by the continuous random variable with probability density function given by $$f ( t ) = \begin{cases} k \sqrt { } t & 1 \leqslant t \leqslant 4
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 14 }\).
2. Find the mean time taken for clothes to dry.
3. Find the median time taken for clothes to dry.
4. Find the probability that the time taken for clothes to dry is between the mean time and the median time.

5 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \sqrt { } 2\).
2. Find \(\mathrm { P } ( X > 0.4 )\).
3. Find the upper quartile of \(X\).
4. Find the probability that exactly 3 out of 5 random observations of \(X\) have values greater than the upper quartile.

4 The random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - x } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { \mathrm { e } } { \mathrm { e } - 1 }\).
2. Find \(\mathrm { E } ( X )\) in terms of e.

4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 2 } { 3 }\).
2. Find the median of \(X\).

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \left( 4 - x ^ { 2 } \right) & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 32 }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\) and hence write down the value of \(\mathrm { E } ( X )\).
3. Find \(\mathrm { P } ( X < 1 )\).

7 A continuous random variable \(X _ { 1 }\) has probability density function given by $$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
4. Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
5. The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).

7 The time, in minutes, for which a customer is prepared to wait on a telephone complaints line is modelled by the random variable \(X\). The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 81 }\).
2. Find \(\mathrm { E } ( X )\).
3. (a) Show that the value \(y\) which satisfies \(\mathrm { P } ( X < y ) = \frac { 3 } { 5 }\) satisfies $$5 y ^ { 4 } - 90 y ^ { 2 } + 243 = 0 .$$ (b) Using the substitution \(w = y ^ { 2 }\), or otherwise, solve the equation in part (a) to find the value of \(y\).

3 The continuous random variable \(T\) has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 0 ,
\frac { a } { \mathrm { e } } & 0 \leqslant t < 2 ,
a \mathrm { e } ^ { - \frac { 1 } { 2 } t } & t \geqslant 2 , \end{cases}$$ where \(a\) is a positive constant.

1. Show that \(a = \frac { 1 } { 4 } \mathrm { e }\).
2. Find the upper quartile of \(T\).

1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} a & 0 \leqslant x \leqslant 1 ,
\frac { a } { x ^ { 2 } } & x > 1 ,
0 & \text { otherwise. } \end{cases}$$ Find the value of the constant \(a\).

4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k x & 0 \leqslant x \leqslant a
0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.

1. State what the letter \(x\) represents.
2. Find \(k\) in terms of \(a\).
3. Find \(\operatorname { Var } ( X )\) in terms of \(a\).

8 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k x ^ { - a } & x \geqslant 1
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants and \(a\) is greater than 1 .

1. Show that \(k = a - 1\).
2. Find the variance of \(X\) in the case \(a = 4\).
3. It is given that \(\mathrm { P } ( X < 2 ) = 0.9\). Find the value of \(a\), correct to 3 significant figures.

1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 2 } { 5 } & - a \leqslant x < 0
\frac { 2 } { 5 } \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$ Find

1. the value of the constant \(a\),
2. \(\mathrm { E } ( X )\).

3 The random variable \(X\) has the following probability density function: $$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1
0 & \text { elsewhere } \end{cases}$$ where \(k\) is a positive constant.

1. Calculate the value of \(k\).
2. Sketch the probability density function.
3. Calculate \(\operatorname { Var } ( X )\).
4. Find a cubic equation satisfied by the upper quartile \(q\), and hence verify that \(q = 0.35\) to 2 decimal places.
5. A random sample of 40 values of \(X\) is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.

1 The random variable \(T\) has probability density defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { t } { 8 } & 0 \leq t \leq k
0 & \text { otherwise } \end{array} \right.$$ Find the value of \(k\)
[0pt] [1 mark] $$\begin{array} { l l l l } \frac { 1 } { 16 } & \frac { 1 } { 4 } & 4 & 16 \end{array}$$

7. The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ( 5 - x ) , & 0 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 56 }\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Evaluate \(\mathrm { E } ( X )\).
4. Find the modal value of \(X\).
5. Verify that the median value of \(X\) lies between 2.3 and 2.5.
6. Comment on the skewness of \(X\). Justify your answer. \section*{END}

7. The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4
0 , & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 2 } { 9 }\). Find
2. \(\mathrm { E } ( X )\),
3. the mode of \(X\).
4. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
5. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
6. Deduce the value of the median and comment on the shape of the distribution.

7. In a competition at a funfair, participants have to stay on a log being rotated in a pool of water for as long as possible. The length of time, in tens of seconds, that the competitors stay on the log is modelled by the random variable \(T\) with the following probability density function: $$\mathrm { f } ( t ) = \begin{cases} k ( t - 3 ) ^ { 2 } , & 0 \leq t \leq 3
0 , & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 1 } { 9 }\).
2. Sketch f \(( t )\) for all values of \(t\).
3. Show that the mean time that competitors stay on the \(\log\) is 7.5 seconds. When the competition is next run the organisers decide to make it easier at first by spinning the log more slowly and then increasing the speed of rotation. The length of time, in tens of seconds, that the competitors now stay on the log is modelled by the random variable \(S\) with the following probability density function: $$f ( s ) = \begin{cases} \frac { 1 } { 12 } \left( 8 - s ^ { 3 } \right) , & 0 \leq s \leq 2
   0 , & \text { otherwise } \end{cases}$$
4. Find the change in the mean time that competitors stay on the log.

7 Many cars have pollen filters to try to remove as much pollen as possible from the passenger compartment. In a test car, the amount of pollen is regularly monitored. The amount of pollen is measured using a scale from 0 to 1 , and is modelled by the continuous random variable \(X\) with probability density function given by
\(f ( x ) = \begin{cases} k \left( 5 x ^ { 4 } - 16 x ^ { 2 } + 11 x \right) & 0 \leqslant x \leqslant 1 ,
0 & \text { otherwise } , \end{cases}\)
where \(k\) is a positive constant.

1. Show that \(k = \frac { 6 } { 7 }\).
2. Determine \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
3. Verify that the median amount of pollen according to the model lies between 0.417 and 0.418.

5 \begin{figure}[h] \end{figure} The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) and Figure 1 shows a sketch of \(\mathrm { f } ( x )\) where $$f ( x ) = \left\{ \begin{array} { c c } k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi
0 & \text { otherwise } \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 2 \pi }\) The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\)
   The probability density function of \(Y\) is \(g ( y )\), where $$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$ Given that \(\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )\)
2. find the exact value of \(\sigma\)
3. Calculate the error in using \(\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)\) as an approximation to \(\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)\)