5.03a Continuous random variables: pdf and cdf

1. A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that \(5 \%\) of people who buy a ticket do not turn up for the journey.

The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable \(X\) represents the number of these people who do not turn up for the journey.

1. State one assumption required to model \(X\) as a binomial distribution. For this week's journey find,
2. 1. the probability that all 50 people turn up for the journey,
   2. \(\mathrm { P } ( X = 1 )\) The bus company receives \(\pounds 20\) for each ticket sold and all 50 tickets are sold. It must pay out \(\pounds 60\) to each person who buys a ticket and turns up for the journey but does not have a seat.
3. Find the bus company's expected total earnings per journey.
   

3. Figure 1 shows an accurate graph of the cumulative distribution function, \(\mathrm { F } ( x )\), for the continuous random variable \(X\) \begin{figure}[h] \end{figure}

1. Find \(\mathrm { P } ( 3 < X < 7 )\) The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} a & 2 \leqslant x < 4 \\ b & 4 \leqslant x < 6 \\ c & 6 \leqslant x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants.
2. Find the value of \(a\), the value of \(b\) and the value of \(c\)
3. Find \(\mathrm { E } ( X )\)

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) where \(0 < a < b\)

Given that \(\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }\)

1. 1. show that \(\mathrm { E } ( X ) = \frac { 5 a } { 2 }\)
   2. find \(\mathrm { P } ( X > b - 4 a )\) The continuous random variable \(Y\) is uniformly distributed over the interval [3, c] where \(c > 3\) Given that \(\operatorname { Var } ( Y ) = 3 c - 9\), find
2. 1. the value of \(c\)
   2. \(\mathrm { P } ( 2 Y - 7 < 20 - Y )\)
   3. \(\mathrm { E } \left( Y ^ { 2 } \right)\)

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

$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0 \\ \frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.

1. Show that \(c = \frac { 1 } { 12 }\)
2. 1. Sketch the probability density function.
   2. Explain why the mode of \(X = 0\)
3. Find the cumulative distribution function of \(X\), for all values of \(x\)
4. Find, to 3 significant figures, the value of \(d\) such that \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)

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4. A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( a - x ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.

1. Show that \(k a ^ { 3 } = 3\) Given that \(\mathrm { E } ( X ) = 1.5\)
2. use algebraic integration to show that \(a = 6\)
3. Verify that the median of \(X\) is 1.2 to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-15_2255_50_314_34}
   

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5. A piece of wood \(A B\) is 3 metres long. The wood is cut at random at a point \(C\) and the random variable \(W\) represents the length of the piece of wood \(A C\).

1. Find the probability that the length of the piece of wood \(A C\) is more than 1.8 metres. The two pieces of wood \(A C\) and \(C B\) form the two shortest sides of a right-angled triangle. The random variable \(X\) represents the length of the longest side of the right-angled triangle.
2. Show that \(X ^ { 2 } = 2 W ^ { 2 } - 6 W + 9\) [0pt] [You may assume for random variables \(S , T\) and \(U\) and for constants \(a\) and \(b\) that if \(S = a T + b U\) then \(\mathrm { E } ( S ) = a \mathrm { E } ( T ) + b \mathrm { E } ( U ) ]\)
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. Find \(\mathrm { P } \left( X ^ { 2 } > 5 \right)\)

2 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < - k \\ \frac { x + k } { 4 k } & - k \leqslant x \leqslant 3 k \\ 1 & x > 3 k \end{array} \right.$$ where \(k\) is a positive constant.

1. Specify fully, in terms of \(k\), the probability density function of \(X\)
2. Write down, in terms of \(k\), the value of \(\mathrm { E } ( X )\)
3. Show that \(\operatorname { Var } ( X ) = \frac { 4 } { 3 } k ^ { 2 }\)
4. Find, in terms of \(k\), the value of \(\mathrm { E } \left( 3 X ^ { 2 } \right)\)

4 The continuous random variable \(X\) has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3 \\ k & 3 < x \leqslant 6 \\ \frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\)
2. Show that \(k = \frac { 1 } { 6 }\)
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 61 } { 12 }\)
4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
5. Describe the skewness of the distribution, giving a reason for your answer.
   

7 The sides of a square are each of length \(L \mathrm {~cm}\) and its area is \(A \mathrm {~cm} ^ { 2 }\) Given that \(A\) is uniformly distributed on the interval [10,30]

1. find \(\mathrm { P } ( L \geqslant 4.5 )\)
2. find \(\operatorname { Var } ( L )\)
   \includegraphics[max width=\textwidth, alt={}]{a009b02e-4cd3-497b-a141-4630c653e20b-28_2655_1947_114_116}

The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\), shown in the diagram, where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{f4fa6add-5860-4c88-bb70-f3edd9b22211-12_511_1096_351_351}

1. Find \(\mathrm { P } ( X < 10 k )\)
2. Show that \(k = \frac { 1 } { \pi }\)
3. Find, in terms of \(\pi\), the values of
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\) Circles are drawn with area \(A\), where $$A = \pi \left( X + \frac { 2 } { \pi } \right) ^ { 2 }$$
   (d) Find \(\mathrm { E } ( A )\)

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ a x + b x ^ { 2 } & 0 \leqslant x \leqslant k \\ 1 & x > k \end{array} \right.$$ where \(a , b\) and \(k\) are positive constants.

1. Show that \(a k = 1 - b k ^ { 2 }\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\)
2. show that \(5 b k ^ { 3 } = 36 - 15 k\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\) and \(\operatorname { Var } ( X ) = \frac { 22 } { 75 }\)
3. show that \(5 b k ^ { 4 } = 52 - 10 k ^ { 2 }\) Given that \(k < 3\)
4. find the value of \(k\)
5. Hence find the value of \(a\) and the value of \(b\)

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

$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2 \\ \frac { 3 } { 20 } & 2 < g \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
2. Find \(\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )\) The continuous random variable \(H\) is such that \(\mathrm { E } ( H ) = 12\) and \(\operatorname { Var } ( H ) = 2.4\)
3. Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\) Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.)

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 \(975 \mathrm {~cm} ^ { 2 }\)

1. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ k \left( a x + b x ^ { 3 } - x ^ { 4 } - 4 \right) & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(a\), \(b\) and \(k\) are non-zero constants.
Given that the mode of \(X\) is 1.5

1. show that \(b = 3\)
2. Hence show that \(a = 2\)
3. Show that the median of \(X\) lies between 1.4 and 1.5

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. \(\mathrm { P } ( T < 3.4 )\)

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 \(x\).

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. 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.

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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