5.03f Relate pdf-cdf: medians and percentiles

1. A game uses two turntables, one red and one yellow. Each turntable has a point marked on the circumference that is lined up with an arrow at the start of the game. Jim spins both turntables and measures the distance, in metres, each point is from the arrow, around the circumference in an anticlockwise direction when the turntables stop spinning.

The continuous random variable \(Y\) represents the distance, in metres, the point is from the arrow for the yellow turntable. The cumulative distribution function of \(Y\) is given by \(\mathrm { F } ( y )\) where $$F ( y ) = \left\{ \begin{array} { c r } 0 & y < 0 \\ 1 - \left( \alpha + \beta y ^ { 2 } \right) & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{array} \right.$$

1. Explain why (i) \(\alpha = 1\) $$\text { (ii) } \beta = - \frac { 1 } { 25 }$$
2. Find the probability density function of \(Y\) The continuous random variable \(R\) represents the distance, in metres, the point is from the arrow for the red turntable. The distribution of \(R\) is modelled by a continuous uniform distribution over the interval \([ d , 3 d ]\) Given that \(\mathrm { P } \left( R > \frac { 11 } { 5 } \right) = \mathrm { P } \left( Y > \frac { 5 } { 3 } \right)\)
3. find the value of \(d\) In the game each turntable is spun 3 times. The distance between the point and the arrow is determined for each spin. To win a prize, at least 5 of the distances the point is from the arrow when a turntable is spun must be less than \(\frac { 11 } { 5 } \mathrm {~m}\) Jo plays the game once.
4. Calculate the probability of Jo winning a prize.

1. A point is to be randomly plotted on the \(x\)-axis, where the units are measured in cm .

The random variable \(R\) represents the \(x\) coordinate of the point on the \(x\)-axis and \(R\) is uniformly distributed over the interval [-5,19] A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin.

1. Find the exact probability that the point is plotted to the right of the origin.
2. Find the exact probability that the point is plotted more than 3.5 cm away from the origin.
3. Sketch the cumulative distribution function of \(R\) Three independent points with \(x\) coordinates \(R _ { 1 } , R _ { 2 }\) and \(R _ { 3 }\) are plotted on the \(x\)-axis.
4. Find the exact probability that
   1. all three points are more than 10 cm from the origin
   2. the point furthest from the origin is more than 10 cm from the origin.

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

$$f ( x ) = \begin{cases} 0.1 x & 0 \leqslant x < 2 \\ k x ( 8 - x ) & 2 \leqslant x < 4 \\ a & 4 \leqslant x < 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants.
It is known that \(\mathrm { P } ( X < 4 ) = \frac { 31 } { 45 }\)

1. Find the exact value of \(k\)
2. 1. Find the exact value of \(a\)
   2. Find the exact value of \(\mathrm { P } ( 0 \leqslant X \leqslant 5.5 )\)
3. Specify fully the cumulative distribution function of \(X\)

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

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 48 } \left( x ^ { 2 } - 8 x + c \right) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that \(c = 31\)
2. Find \(\mathrm { P } ( 2 < X < 3 )\)
3. State whether the lower quartile of \(X\) is less than 3, equal to 3 or greater than 3 Give a reason for your answer. Kei does the following to work out the mode of \(X\) $$\begin{aligned} f ^ { \prime } ( x ) & = \frac { 1 } { 48 } ( 2 x - 8 ) \\ 0 & = \frac { 1 } { 48 } ( 2 x - 8 ) \\ x & = 4 \end{aligned}$$ Hence the mode of \(X\) is 4 Kei's answer for the mode is incorrect.
4. Explain why Kei's method does not give the correct value for the mode.
5. Find the mode of \(X\) Give a reason for your answer.

1. A continuous random variable \(Y\) has cumulative distribution function given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { c r } 0 & y < 3 \\ \frac { 1 } { 16 } \left( y ^ { 2 } - 6 y + a \right) & 3 \leqslant y \leqslant 5 \\ \frac { 1 } { 12 } ( y + b ) & 5 < y \leqslant 9 \\ \frac { 1 } { 12 } \left( 100 y - 5 y ^ { 2 } + c \right) & 9 < y \leqslant 10 \\ 1 & y > 10 \end{array} \right.$$ where \(a\), \(b\) and \(c\) are constants.

1. Find the value of \(a\) and the value of \(c\)
2. Find the value of \(b\)
3. Find \(\mathrm { P } ( 6 < Y \leqslant 9 )\) Show your working clearly.
4. Specify the probability density function, f(y), for \(5 < y \leqslant 9\) Using the information $$\int _ { 3 } ^ { 5 } ( 6 y - 5 ) f ( y ) d y + \int _ { 9 } ^ { 10 } ( 6 y - 5 ) f ( y ) d y = 26.5$$
5. find \(\mathrm { E } ( 6 Y - 5 )\) You should make your method clear.

2 The continuous random variable \(H\) has cumulative distribution function given by $$\mathrm { F } ( h ) = \left\{ \begin{array} { l r } 0 & h \leqslant 0 \\ \frac { h ^ { 2 } } { 48 } & 0 < h \leqslant 4 \\ \frac { h } { 6 } - \frac { 1 } { 3 } & 4 < h \leqslant 5 \\ \frac { 3 } { 10 } h - \frac { h ^ { 2 } } { 75 } - \frac { 2 } { 3 } & 5 < h \leqslant d \\ 1 & h > d \end{array} \right.$$ where \(d\) is a constant.

1. Show that \(2 d ^ { 2 } - 45 d + 250 = 0\)
2. Find \(\mathrm { P } ( H < 1.5 \mid 1 < H < 4.5 )\)
3. Find the probability density function \(\mathrm { f } ( h )\) You may leave the limits of \(h\) in terms of \(d\) where necessary.

1. The lifetime of a particular battery, \(T\) hours, is modelled using the cumulative distribution function

$$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 8 \\ \frac { 1 } { 96 } \left( 74 t - \frac { 5 } { 2 } t ^ { 2 } + k \right) & 8 \leqslant t \leqslant 12 \\ 1 & t > 12 \end{array} \right.$$

1. Show that \(k = - 432\)
2. Find the probability density function of \(T\), for all values of \(t\).
3. Write down the mode of \(T\).
4. Find the median of \(T\).
5. Find the probability that a randomly selected battery has a lifetime of less than 9 hours. A battery is selected at random. Given that its lifetime is at least 9 hours,
6. find the probability that its lifetime is no more than 11 hours.

4. \begin{figure}[h] \end{figure} A continuous random variable \(X\) has the probability density function \(\mathrm { f } ( x )\) shown in Figure 1 $$\mathrm { f } ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 5 \\ k & 5 < x \leqslant 10.5 \\ 0 & \text { otherwise } \end{cases}$$ where \(m\) and \(k\) are constants.

1. 1. Show that \(k = \frac { 1 } { 8 }\)
   2. Find the value of \(m\)
2. Find \(\mathrm { E } ( X )\)
3. Find the interquartile range of \(X\)

3. The function \(\mathrm { f } ( x )\) is defined as $$f ( x ) = \begin{cases} \frac { 1 } { 9 } ( x + 5 ) ( 3 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Albert believes that \(\mathrm { f } ( x )\) is a valid probability density function.

1. Sketch \(\mathrm { f } ( x )\) and comment on Albert's belief. The continuous random variable \(Y\) has probability density function given by $$g ( y ) = \begin{cases} k y \left( 12 - y ^ { 2 } \right) & 1 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
2. Use calculus to find the mode of \(Y\)
3. Use algebraic integration to find the value of \(k\)
4. Find the median of \(Y\) giving your answer to 3 significant figures.
5. Describe the skewness of the distribution of \(Y\) giving a reason for your answer.

1. In the summer Kylie catches a local steam train to work each day. The published arrival time for the train is 10 am.

The random variable \(W\) is the train's actual arrival time minus the published arrival time, in minutes. When the value of \(W\) is positive, the train is late. The cumulative distribution function \(\mathrm { F } ( w )\) is shown in the sketch below. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-06_583_1235_589_349}

1. Specify fully the probability density function \(\mathrm { f } ( w )\) of \(W\).
2. Write down the value of \(\mathrm { E } ( \mathrm { W } )\)
3. Calculate \(\alpha\) such that \(\mathrm { P } ( \alpha \leqslant W \leqslant 1.6 ) = 0.35\) A day is selected at random.
4. Calculate the probability that on this day the train arrives between 1.2 minutes late and 2.4 minutes late. Given that on this day the train was between 1.2 minutes late and 2.4 minutes late,
5. calculate the probability that it was more than 2 minutes late. A random sample of 40 days is taken.
6. Calculate the probability that for at least 10 of these days the train is between 1.2 minutes late and 2.4 minutes late. DO NOT WRITEIN THIS AREA

5. The waiting time, \(T\) minutes, of a customer to be served in a local post office has probability density function $$\mathrm { f } ( t ) = \begin{cases} \frac { 1 } { 50 } ( 18 - 2 t ) & 0 \leqslant t \leqslant 3 \\ \frac { 1 } { 20 } & 3 < t \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ Given that the mean number of minutes a customer waits to be served is 1.66

1. use algebraic integration to find \(\operatorname { Var } ( T )\), giving your answer to 3 significant figures.
2. Find the cumulative distribution function \(\mathrm { F } ( t )\) for all values of \(t\).
3. Calculate the probability that a randomly chosen customer's waiting time will be more than 2 minutes.
4. Calculate \(\mathrm { P } ( [ \mathrm { E } ( T ) - 2 ] < T < [ \mathrm { E } ( T ) + 2 ] )\)

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3. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ 4 a x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ a \left( b x ^ { 3 } - x ^ { 4 } + 1 \right) & 1 < x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$ where \(a\) and \(b\) are positive constants.

1. Show that \(b = 4\)
2. Find the exact value of \(a\)
3. Find \(\mathrm { P } ( X > 2.25 )\)
4. Showing your working clearly,
   1. sketch the probability density function of \(X\)
   2. calculate the mode of \(X\)

6. The continuous random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by $$f ( y ) = \begin{cases} \frac { 1 } { 14 } ( y + 2 ) & - 1 < y \leqslant 1 \\ \frac { 3 } { 14 } & 1 < y \leqslant 3 \\ \frac { 1 } { 14 } ( 6 - y ) & 3 < y \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the probability density function \(\mathrm { f } ( \mathrm { y } )\) Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 131 } { 21 }\)
2. find \(\operatorname { Var } ( 2 Y - 3 )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\)
3. Show that \(\mathrm { F } ( y ) = \frac { 1 } { 14 } \left( \frac { y ^ { 2 } } { 2 } + 2 y + \frac { 3 } { 2 } \right)\) for \(- 1 < y \leqslant 1\)
4. Find \(\mathrm { F } ( y )\) for all values of \(y\)
5. Find the exact value of the 30th percentile of \(Y\)
6. Find \(\mathrm { P } ( 4 Y \leqslant 5 \mid Y \leqslant 3 )\)

1. A random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & - \frac { 1 } { 2 } \leqslant x < \frac { 1 } { 2 } \\ 2 x - \frac { 3 } { 4 } & \frac { 1 } { 2 } \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\)
2. By forming and solving an equation in \(k\), show that \(k = 1.25\)
3. Use calculus to find \(\mathrm { E } ( X )\)
4. Calculate the interquartile range of \(X\)

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 3 \\ \frac { 1 } { 6 } ( x - 3 ) ^ { 2 } & 3 \leqslant x < 4 \\ \frac { x } { 3 } - \frac { 7 } { 6 } & 4 \leqslant x < c \\ 1 - \frac { 1 } { 6 } ( d - x ) ^ { 2 } & c \leqslant x < 7 \\ 1 & x \geqslant 7 \end{array} \right.$$ where \(c\) and \(d\) are constants.

1. Show that \(c = 6\)
2. Find \(\mathrm { P } ( X > 3.5 )\)
3. Find \(\mathrm { P } ( X > 4.5 \mid 3.5 < X < 5.5 )\)

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$\mathrm { f } ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant d \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\), \(c\) and \(d\) are constants such that

• \(b x + c = a x ^ { 3 }\) at \(x = 4\)
• \(b x + c\) is a straight line segment with end coordinates ( \(4,64 a\) ) and ( \(d , 0\) )
  1. State the mode of \(X\)

Given that the mode of \(X\) is equal to the median of \(X\)

use algebraic integration to show that \(a = \frac { 1 } { 128 }\)

Find the value of \(d\)

Hence find the value of \(b\) and the value of \(c\)

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

$$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)

2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$

1. Find \(\mathrm { P } ( X > 4 )\)
2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
3. Find the probability density function of \(X\), specifying it for all values of \(x\)
4. Write down the value of \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\)
6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)

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. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0 \\ k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2 \\ 8 k , & x > 2 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 8 }\).
2. Find the median of \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
5. Write down the mode of \(X\).
6. Find \(\mathrm { E } ( X )\).
7. Comment on the skewness of this distribution.

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.

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.

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

1. Find \(\mathrm { P } ( X > 0.3 )\).
2. Verify that the median value of \(X\) lies between \(x = 0.59\) and \(x = 0.60\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Evaluate \(\mathrm { E } ( X )\).
5. Find the mode of \(X\).
6. Comment on the skewness of \(X\). Justify your answer.

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$f ( x ) = \left\{ \begin{array} { c c } 2 ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the median of \(X\).
5. Comment on the skewness of this distribution. Give a reason for your answer.