5.03e Find cdf: by integration

6. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < 0 \\ \frac { x ^ { 2 } } { 20 } ( 9 - 2 x ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$

1. Verify that the median of \(X\) lies between 1.23 and 1.24
2. Specify fully the probability density function \(\mathrm { f } ( x )\).
3. Find the mode of \(X\).
4. Describe the skewness of this distribution. Justify your answer.

1. A continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 2 \\ \frac { 1 } { 20 } \left( x ^ { 2 } - 4 \right) & 2 \leqslant x \leqslant 4 \\ \frac { 1 } { 5 } ( 2 x - 5 ) & 4 < x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X > 4 )\)
2. Find the probability density function of \(X\), specifying it for all values of \(x\).
3. Find the value of \(a\) such that \(\mathrm { P } ( 3 < X < a ) = 0.642\)
4. Find the probability density function of \(X\), specifying it for all values of \(x\).

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

$$f ( x ) = \begin{cases} \frac { 2 x } { 15 } & 0 \leqslant x \leqslant k \\ \frac { 1 } { 5 } ( 5 - x ) & k < x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Showing your working clearly, find the value of \(k\).
2. Write down the mode of \(X\).
3. Find \(\mathrm { P } \left( \left. X \leqslant \frac { k } { 2 } \right\rvert \, X \leqslant k \right)\)

1. The waiting times, in minutes, between flight take-offs at an airport are modelled by the continuous random variable \(X\) with probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 2 \leqslant x \leqslant 7 \\ 0 & \text { otherwise } \end{cases}$$

1. Write down the name of this distribution. A randomly selected flight takes off at 9am
2. Find the probability that the next flight takes off before 9.05 am
3. Find the probability that at least 1 of the next 5 flights has a waiting time of more than 6 minutes.
4. Find the cumulative distribution function of \(X\), for all \(x\)
5. Sketch the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 7\) On foggy days, an extra 2 minutes is added to each waiting time.
6. Find the mean and variance of the waiting times between flight take-offs on foggy days.

6. A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} a x - b x ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ Given that the mode is 1

1. show that \(a = 2 b\)
2. Find the value of \(a\) and the value of \(b\)
3. Calculate F(1.5)
4. State whether the upper quartile of \(X\) is greater than 1.5, equal to 1.5, or less than 1.5 Give a reason for your answer.
   

3. The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} a x + b & 1 \leqslant x < 4 \\ \frac { 3 } { 2 } - \frac { 1 } { 4 } x & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ as shown in Figure 1, where \(a\) and \(b\) are constants. \begin{figure}[h] \end{figure}

1. Show that the median of \(X\) is 4
2. Find the value of \(a\) and the value of \(b\)
3. Specify fully the cumulative distribution function of \(X\)

5. A call centre records the length of time, \(T\) minutes, its customers wait before being connected to an agent. The random variable \(T\) has a cumulative distribution function given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { l r } 0 & t < 0 \\ 0.3 t - 0.004 t ^ { 3 } & 0 \leqslant t \leqslant 5 \\ 1 & t > 5 \end{array} \right.$$

1. Find the proportion of customers waiting more than 4 minutes to be connected to an agent.
2. Given that a customer waits more than 2 minutes to be connected to an agent, find the probability that the customer waits more than 4 minutes.
3. Show that the upper quartile lies between 2.7 and 2.8 minutes.
4. Find the mean length of time a customer waits to be connected to an agent.

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.

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. 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.

6 In this question solutions relying entirely on calculator technology are not acceptable.
The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(4 a + 4 b = 1\) Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 17 } { 5 }\)
2. 1. find an equation in terms of \(a\) only
   2. hence show that \(b = 0.1\)
3. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\)
4. Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.8\)

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.

5. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 3 \\ k \left( y ^ { 2 } - 2 y - 3 \right) & 3 \leqslant y \leqslant \alpha \\ 4 k ( 2 y - 7 ) & \alpha < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$ where \(k\) and \(\alpha\) are constants.

1. Find \(\mathrm { P } ( 4.5 < Y \leqslant 5.5 )\)
2. Find the probability density function \(\mathrm { f } ( \mathrm { y } )\)

5. The random variable \(X\) has cumulative distribution function given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ \frac { 1 } { 100 } \left( a x ^ { 3 } + b x ^ { 2 } + 15 x \right) & 0 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$ Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 6.25\)

1. show that \(6 a + b = 0\)
2. find the value of \(a\) and the value of \(b\)
3. find \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\)

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. 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 \(Y\) 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.

4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable \(X\) represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of \(X\) and specify it fully.
2. Calculate the mean time of arrival of the bus.
3. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
4. Find the probability that Jean is late for work.