5.03c Calculate mean/variance: by integration

1. The random variable \(W\) has a continuous uniform distribution over the interval \([ - 6 , a ]\) where \(a\) is a constant.

Given that \(\operatorname { Var } ( W ) = 27\)

1. show that \(a = 12\) Given that \(\mathrm { P } ( W > b ) = \frac { 3 } { 5 }\)
2. 1. find the value of \(b\)
   2. find \(\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)\) A piece of wood \(A B\) has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.
3. Calculate the probability that the area of the rectangle is greater than \(975 \mathrm {~cm} ^ { 2 }\)

2. The amount of flour used by a factory in a week is \(Y\) thousand kg where \(Y\) has probability density function $$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k \left( 4 - y ^ { 2 } \right) & 0 \leqslant y \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the value of \(k\) is \(\frac { 3 } { 16 }\) Use algebraic integration to find
2. the mean number of kilograms of flour used by the factory in a week,
3. the standard deviation of the number of kilograms of flour used by the factory in a week,
4. the probability that more than 1500 kg of flour will be used by the factory next week.

1. The continuous random variable \(T\) is uniformly distributed on the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\)

Given that \(\mathrm { E } ( T ) = 2\) and \(\operatorname { Var } ( T ) = \frac { 16 } { 3 }\), find

1. the value of \(\alpha\) and the value of \(\beta\),
2. \(\mathrm { P } ( T < 3.4 )\)

A piece of spaghetti has length \(2 c\), where \(c\) is a positive constant. It is cut into two pieces at a random point. The continuous random variable \(X\) represents the length of the longer piece and is uniformly distributed over the interval \([ c , 2 c ]\).

1. Sketch the graph of the probability density function of \(X\)
2. Use integration to prove that \(\operatorname { Var } ( X ) = \frac { c ^ { 2 } } { 12 }\)
3. Find the probability that the longer piece is more than twice the length of the shorter piece.

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.

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.

7. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)

1. Find an expression, in terms of \(a\) and \(b\), for \(\mathrm { E } ( 3 - 2 X )\)
2. Find \(\mathrm { P } \left( X > \frac { 1 } { 3 } b + \frac { 2 } { 3 } a \right)\) Given that \(\mathrm { E } ( X ) = 0\)
3. find an expression, in terms of \(b\) only, for \(\mathrm { E } \left( 3 X ^ { 2 } \right)\) Given also that the range of \(X\) is 18
4. find \(\operatorname { Var } ( X )\)
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3. A machine pours oil into bottles. It is electronically controlled to cut off the flow of oil randomly between 100 ml and \(k \mathrm { ml }\), where \(k > 100\). It is equally likely to cut off the flow at any point in this range. The random variable \(X\) is the volume of oil poured into a bottle. Given that \(\mathrm { P } ( 102 \leqslant X \leqslant k ) = \frac { 2 } { 3 }\)

1. show that \(k = 106\)
2. Find the probability that the volume of oil poured into a bottle is
   1. less than 105 ml ,
   2. exactly 105 ml .
3. Write down the value of \(\mathrm { E } ( X )\)
4. Find the 15th percentile of this distribution.
5. Determine the value of \(x\) such that \(3 \mathrm { P } ( X \leqslant x - 1.5 ) = \mathrm { P } ( X \geqslant x + 1.5 )\)

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. The continuous random variable \(Y\) has the following probability density function

$$f ( y ) = \begin{cases} \frac { 6 } { 25 } ( y - 1 ) & 1 \leqslant y < 2 \\ \frac { 3 } { 50 } \left( 4 y ^ { 2 } - y ^ { 3 } \right) & 2 \leqslant y < 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch f(y)
2. Find the mode of \(Y\)
3. Use algebraic integration to calculate \(\mathrm { E } \left( Y ^ { 2 } \right)\) Given that \(\mathrm { E } ( Y ) = 2.696\)
4. find \(\operatorname { Var } ( Y )\)
5. Find the value of \(y\) for which \(\mathrm { P } ( Y \geqslant y ) = 0.9\) Give your answer to 3 significant figures.

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 \(\mathrm { P } ( X \geqslant k ) = 0.8\)

2. The weekly sales, \(S\), in thousands of pounds, of a small business has probability density function $$\mathrm { f } ( s ) = \left\{ \begin{array} { c c } k ( s - 2 ) ( 10 - s ) & 2 < s < 10 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to show that \(k = \frac { 3 } { 256 }\)
2. Write down the value of \(\mathrm { E } ( S )\)
3. Use algebraic integration to find the standard deviation of the weekly sales. A week is selected at random.
4. Showing your working, find the probability that this week's sales exceed \(\pounds 7100\) Give your answer to one decimal place. A quarter is defined as 12 consecutive weeks. The discrete random variable \(X\) is the number of weeks in a quarter in which the weekly sales exceed £7100 The manager earns a bonus at the following rates:

   \(\boldsymbol { X }\)	Bonus Earned
   \(X \leqslant 5\)	\(\pounds 0\)
   \(X = 6\)	\(\pounds 1000\)
   \(X \geqslant 7\)	\(\pounds 5000\)

5. Using your answer to part (d), calculate the manager's expected bonus per quarter.

4. In a computer game, a ship appears randomly on a rectangular screen. The continuous random variable \(X \mathrm {~cm}\) is the distance of the centre of the ship from the bottom of the screen. The random variable \(X\) is uniformly distributed over the interval \([ 0 , \alpha ]\) where \(\alpha \mathrm { cm }\) is the height of the screen. Given that \(\mathrm { P } ( X > 6 ) = 0.6\)

1. find the value of \(\alpha\)
2. find \(\mathrm { P } ( 4 < X < 10 )\) The continuous random variable \(Y\) cm is the distance of the centre of the ship from the left-hand side of the screen. The random variable \(Y\) is uniformly distributed over the interval [ 0,20 ] where 20 cm is the width of the screen.
3. Find the mean and the standard deviation of \(Y\).
4. Find \(\mathrm { P } ( | Y - 4 | < 2 )\)
5. Given that \(X\) and \(Y\) are independent, find the probability that the centre of the ship appears
   1. in a square of side 4 cm which is at the centre of the screen,
   2. within 5 cm of a side or the top or the bottom of the screen.

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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2. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 8 < X < 14 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 11\)

1. write down \(\mathrm { P } ( X > 14 )\)
2. find \(\mathrm { P } ( 6 X > a + b )\) (ii) Susie makes a strip of pasta 45 cm long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable \(S\) is the length of the shortest piece of pasta.
   1. Write down the distribution of \(S\)
   2. Calculate the probability that the shortest piece of pasta is less than 12 cm long. Susie makes 20 strips of pasta, all 45 cm long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces.
   3. Calculate the probability that exactly 6 of the pieces of pasta are less than 12 cm long.

1. A bag contains a large number of counters.

40\% of the counters are numbered 1 \(35 \%\) of the counters are numbered 2 \(25 \%\) of the counters are numbered 3 In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.

1. Show that Alif gets a score of 8 with probability 0.0875
2. Find the sampling distribution of Alif's score.
3. Calculate Alif's expected score.

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 \(\mathrm { P } ( 4 Y \leqslant 5 \mid Y \leqslant 3 )\)

Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.

1. Find the probability that in 15 metres of her cloth there are
   1. less than 3 faults,
   2. at least 6 faults. Each roll contains 100 metres of Bhavna's cloth.
      She selects 15 rolls at random.
2. Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
   Each piece of her cloth is 4 metres long.
   The cost to make each piece is \(\pounds 5.00\) She sells each piece of her cloth that contains no faults for \(\pounds 7.40\) She sells each piece of her cloth that contains faults for \(\pounds 2.00\)
3. Find the expected profit that Bhavna will make on each piece of her cloth that she sells.

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 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. Every morning Navtej travels from home to work. Navtej leaves home at a random time between 08:00 and 08:15

• It always takes Navtej 3 minutes to walk to the bus stop
• Buses run every 15 minutes and Navtej catches the first bus that arrives
• Once Navtej has caught the bus it always takes a further 29 minutes for Navtej to reach work

The total time, \(T\) minutes, for Navtej's journey from home to work is modelled by a continuous uniform distribution over the interval \([ \alpha , \beta ]\)

1. 1. Show that \(\alpha = 32\)
   2. Show that \(\beta = 47\)
2. State fully the probability density function for this distribution.
3. Find the value of
   1. \(\mathrm { E } ( T )\)
   2. \(\operatorname { Var } ( T )\)
4. Find the probability that the time for Navtej's journey is within 5 minutes of 35 minutes.

1. A manufacturer makes t -shirts in 3 sizes, small, medium and large.

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts.

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 \(\mathrm { E } ( Y )\)