5.03a Continuous random variables: pdf and cdf

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.

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. The time, in minutes, spent waiting for a call to a call centre to be answered is modelled by the random variable \(T\) with probability density function

$$f ( t ) = \left\{ \begin{array} { l c } \frac { 1 } { 192 } \left( t ^ { 3 } - 48 t + 128 \right) & 0 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to find, in minutes and seconds, the mean waiting time.
2. Show that \(\mathrm { P } ( 1 < T < 3 ) = \frac { 7 } { 16 }\) A supervisor randomly selects 256 calls to the call centre.
3. Use a suitable approximation to find the probability that more than 125 of these calls take between 1 and 3 minutes to be answered.

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.

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.
   

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 \(X\)

A string of length 40 cm is cut into 2 pieces at a random point. The continuous random variable \(L\) represents the length of the longer piece of string.

1. Write down the distribution of \(L\)
2. Find the probability that the length of the longer piece of string is 28 cm to the nearest cm Each piece of string is used to form the perimeter of a square.
3. Calculate the probability that the area of the larger square is less than \(64 \mathrm {~cm} ^ { 2 }\)
4. Calculate the probability that the difference in area between the two squares is greater than \(81 \mathrm {~cm} ^ { 2 }\)

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 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 \(\mathrm { f } ( \mathrm { y } )\)

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.

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 \(\mathrm { P } ( 3 \leqslant X \leqslant 7 )\)

One side of a square is measured to the nearest centimetre and this measurement is multiplied by 4 to estimate the perimeter of the square. The random variable, \(W \mathrm {~cm}\), represents the estimated perimeter of the square minus the true perimeter of the square. \(W\) is uniformly distributed over the interval \([ a , b ]\)

1. Explain why \(a = - 2\) and \(b = 2\) The standard deviation of \(W\) is \(\sigma\)
2. 1. Find \(\sigma\)
   2. Find the probability that the estimated perimeter of the square is within \(\sigma\) of the true perimeter of the square. One side of each of 100 squares are now measured. Using a suitable approximation,
3. find the probability that \(W\) is greater than 1.9 for at least 5 of these squares.

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1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). For \(1 \leqslant x \leqslant 2 , \mathrm { f } ( x )\) is represented by a curve with equation \(\mathrm { f } ( x ) = k \left( \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + a x + 1 \right)\) where \(k\) and \(a\) are constants. For all other values of \(x , \mathrm { f } ( x ) = 0\)

1. Use algebraic integration to show that \(k ( 12 a - 33 ) = 8\) Given that \(a = 5\)
2. calculate the mode of \(X\).

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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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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.

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