5.03a Continuous random variables: pdf and cdf

1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].

Find

1. Elaine's expected checkout time,
2. the variance of the time taken to checkout at the supermarket,
3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
4. find the probability that she will take a total of less than 6 minutes to checkout.
   

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

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that \(k = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find \(\mathrm { P } ( 0.5 < X < 1.5 )\).
5. Write down the median of \(X\) and the mode of \(X\).
6. Describe the skewness of the distribution of \(X\). Give a reason for your answer.

4. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 4,6 ]\).

1. Write down the mean of \(X\).
2. Find \(\mathrm { P } ( X \leqslant 2.4 )\)
3. Find \(\mathrm { P } ( - 3 < X - 5 < 3 )\) The continuous random variable \(Y\) is uniformly distributed over the interval \([ a , 4 a ]\).
4. Use integration to show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }\)
5. Find \(\operatorname { Var } ( Y )\).
6. Given that \(\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)\), find the value of \(a\).

5. The continuous random variable \(T\) is used to model the number of days, \(t\), a mosquito survives after hatching. The probability that the mosquito survives for more than \(t\) days is $$\frac { 225 } { ( t + 15 ) ^ { 2 } } , \quad t \geqslant 0$$

1. Show that the cumulative distribution function of \(T\) is given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \frac { 225 } { ( t + 15 ) ^ { 2 } } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the probability that a randomly selected mosquito will die within 3 days of hatching.
3. Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days. A large number of mosquitoes hatch on the same day.
4. Find the number of days after which only \(10 \%\) of these mosquitoes are expected to survive.

7. The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(10 a + 25 b = 2\) Given that \(\mathrm { E } ( X ) = \frac { 35 } { 12 }\)
2. find a second equation in \(a\) and \(b\),
3. hence find the value of \(a\) and the value of \(b\).
4. Find, to 3 significant figures, the median of \(X\).
5. Comment on the skewness. Give a reason for your answer.

6. The continuous random variable X has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 1 \\ \frac { 1 } { 27 } \left( - x ^ { 3 } + 6 x ^ { 2 } - 5 \right) , & 1 \leq x \leq 4 \\ 1 , & x > 4 \end{array} \right.$$

1. Find the probability density function \(\mathrm { f } ( x )\).
2. Find the mode of \(X\).
3. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
4. Find the mean \(\mu\) of X .
5. Show that \(\mathrm { F } ( \mu ) > 0.5\).
6. Show that the median of \(X\) lies between the mode and the mean.

7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).

1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
   (4) END

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).

1. Write down the probability density function \(\mathrm { f } ( x )\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( X )\),
4. the cumulative distribution function of \(X\), for all \(x\),
5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).

6. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.

1. Show that \(k = \frac { 1 } { 4 }\). Find
2. \(\mathrm { E } ( X )\),
3. the mode of \(X\),
4. the median of \(X\).
5. Comment on the skewness of the distribution.
6. Sketch f(x).

6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).

1. Find \(\mathrm { P } ( A > 3 )\).
2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
3. Find the probability density function of \(Y\).
4. Sketch the probability density function of \(Y\).
5. Write down the mode of \(Y\).
6. Find \(\mathrm { E } ( Y )\).
7. Find \(\mathrm { P } ( Y > 3 )\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\). The part of the sketch from \(x = 0\) to \(x = 4\) consists of an isosceles triangle with maximum at ( \(2,0.5\) ).

1. Write down \(\mathrm { E } ( X )\). The probability density function \(\mathrm { f } ( x )\) can be written in the following form. $$f ( x ) = \begin{cases} a x & 0 \leqslant x < 2 \\ b - a x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the values of the constants \(a\) and \(b\).
3. Show that \(\sigma\), the standard deviation of \(X\), is 0.816 to 3 decimal places.
4. Find the lower quartile of \(X\).
5. State, giving a reason, whether \(\mathrm { P } ( 2 - \sigma < X < 2 + \sigma )\) is more or less than 0.5

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the probability density function \(\mathrm { f } ( x )\) of the random variable \(X\).
For \(0 \leqslant x \leqslant 3 , \mathrm { f } ( x )\) is represented by a curve \(O B\) with equation \(\mathrm { f } ( x ) = k x ^ { 2 }\), where \(k\) is a constant. For \(3 \leqslant x \leqslant a\), where \(a\) is a constant, \(\mathrm { f } ( x )\) is represented by a straight line passing through \(B\) and the point ( \(a , 0\) ). For all other values of \(x , \mathrm { f } ( x ) = 0\).
Given that the mode of \(X =\) the median of \(X\), find

1. the mode,
2. the value of \(k\),
3. the value of \(a\). Without calculating \(\mathrm { E } ( X )\) and with reference to the skewness of the distribution
4. state, giving your reason, whether \(\mathrm { E } ( X ) < 3 , \mathrm { E } ( X ) = 3\) or \(\mathrm { E } ( X ) > 3\).

1. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].

A stick is selected at random from the box.

1. Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
2. Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
3. Find the probability of winning a soft toy.
   

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

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 3 } { 32 } ( x - 1 ) ( 5 - x ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch \(\mathrm { f } ( x )\) showing clearly the points where it meets the \(x\)-axis.
2. Write down the value of the mean, \(\mu\), of \(X\).
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = 9.8\)
4. Find the standard deviation, \(\sigma\), of \(X\). The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 32 } \left( a - 15 x + 9 x ^ { 2 } - x ^ { 3 } \right) & 1 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$ where \(a\) is a constant.
5. Find the value of \(a\).
6. Show that the lower quartile of \(X , q _ { 1 }\), lies between 2.29 and 2.31
7. Hence find the upper quartile of \(X\), giving your answer to 1 decimal place.
8. Find, to 2 decimal places, the value of \(k\) so that $$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$

A manufacturer produces sweets of length \(L \mathrm {~mm}\) where \(L\) has a continuous uniform distribution with range [15, 30].

1. Find the probability that a randomly selected sweet has a length greater than 24 mm . These sweets are randomly packed in bags of 20 sweets.
2. Find the probability that a randomly selected bag will contain at least 8 sweets with length greater than 24 mm .
3. Find the probability that 2 randomly selected bags will both contain at least 8 sweets with length greater than 24 mm .
   

1. The queueing time, \(X\) minutes, of a customer at a till of a supermarket has probability density function

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 3 } { 32 } x ( k - x ) & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the value of \(k\) is 4
2. Write down the value of \(\mathrm { E } ( X )\).
3. Calculate \(\operatorname { Var } ( X )\).
4. Find the probability that a randomly chosen customer's queueing time will differ from the mean by at least half a minute.
   

7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { x ^ { 2 } } { 45 } & 0 \leqslant x \leqslant 3 \\ \frac { 1 } { 5 } & 3 < x < 4 \\ \frac { 1 } { 3 } - \frac { x } { 30 } & 4 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{array} . \right.$$

1. Sketch \(\mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 10\)
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Find \(\mathrm { P } ( X \leqslant 8 )\).

2. The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < 0 \\ \frac { 1 } { 4 } \left( y ^ { 3 } - 4 y ^ { 2 } + k y \right) & 0 \leqslant y \leqslant 2 \\ 1 & y > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability density function of \(Y\), specifying it for all values of \(y\).
3. Find \(\mathrm { P } ( Y > 1 )\).

4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( 3 + 2 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\)
2. Find the mode of \(X\).
3. Use algebraic integration to find \(\mathrm { E } ( X )\). By comparing your answers to parts (b) and (c),
4. describe the skewness of \(X\), giving a reason for your answer.

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\).
2. Show that \(\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leqslant x \leqslant 2\)
3. Use integration to find \(\mathrm { E } ( X )\).
4. Show that the lower quartile of \(X\) lies between 1.425 and 1.435

4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } 3 k & 0 \leqslant x < 1 \\ k x ( 4 - x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Sketch f (x).
2. Write down the mode of \(X\). Given that \(\mathrm { E } ( X ) = \frac { 29 } { 16 }\)
3. describe, giving a reason, the skewness of the distribution.
4. Use integration to find the value of \(k\).
5. Write down the lower quartile of \(X\). Given also that \(\mathrm { P } ( 2 < X < 3 ) = \frac { 11 } { 36 }\)
6. find the exact value of \(\mathrm { P } ( X > 3 )\).

6. In an experiment some children were asked to estimate the position of the centre of a circle. The random variable \(D\) represents the distance, in centimetres, between the child's estimate and the actual position of the centre of the circle. The cumulative distribution function of \(D\) is given by $$\mathrm { F } ( d ) = \left\{ \begin{array} { c c } 0 & d < 0 \\ \frac { d ^ { 2 } } { 2 } - \frac { d ^ { 4 } } { 16 } & 0 \leqslant d \leqslant 2 \\ 1 & d > 2 \end{array} \right.$$

1. Find the median of \(D\).
2. Find the mode of \(D\). Justify your answer. The experiment is conducted on 80 children.
3. Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle.

2. The length of time, in minutes, that a customer queues in a Post Office is a random variable, \(T\), with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } c \left( 81 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(c\) is a constant.

1. Show that the value of \(c\) is \(\frac { 1 } { 486 }\)
2. Show that the cumulative distribution function \(\mathrm { F } ( t )\) is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c c } 0 & t < 0 \\ \frac { t } { 6 } - \frac { t ^ { 3 } } { 1458 } & 0 \leqslant t \leqslant 9 \\ 1 & t > 9 \end{array} \right.$$
3. Find the probability that a customer will queue for longer than 3 minutes. A customer has been queueing for 3 minutes.
4. Find the probability that this customer will be queueing for at least 7 minutes. Three customers are selected at random.
5. Find the probability that exactly 2 of them had to queue for longer than 3 minutes.

6. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 2 x } { 9 } & 0 \leqslant x \leqslant 1 \\ \frac { 2 } { 9 } & 1 < x < 4 \\ \frac { 2 } { 3 } - \frac { x } { 9 } & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find \(\mathrm { E } ( X )\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) for all values of \(x\).
3. Find the median of \(X\).
4. Describe the skewness. Give a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{caaa0133-5b13-4ca7-8e65-8543327c33fd-12_104_61_2412_1884}

4. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 2 \\ k \left( a x + b x ^ { 2 } - x ^ { 3 } \right) & 2 \leqslant x \leqslant 3 \\ 1 & x > 3 \end{array} \right.$$ Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)

1. show that \(b = 8\)
2. find the value of \(k\).