5.03e Find cdf: by integration

7. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0 \\ k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2 \\ 8 k , & x > 2 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 8 }\).
2. Find the median of \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
5. Write down the mode of \(X\).
6. Find \(\mathrm { E } ( X )\).
7. Comment on the skewness of this distribution.

4. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0 \\ \frac { 1 } { 3 } x ^ { 2 } \left( 4 - x ^ { 2 } \right) , & 0 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$$

1. Find \(\mathrm { P } ( X > 0.7 )\).
2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
3. Calculate \(\mathrm { E } ( X )\) and show that, to 3 decimal places, \(\operatorname { Var } ( X ) = 0.057\). One measure of skewness is $$\frac { \text { Mean - Mode } } { \text { Standard deviation } } .$$
4. Evaluate the skewness of the distribution of \(X\).

7. The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4 \\ 0 , & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 2 } { 9 }\). Find
2. \(\mathrm { E } ( X )\),
3. the mode of \(X\).
4. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
5. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
6. Deduce the value of the median and comment on the shape of the distribution.

5. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k x ( x - 2 ) , & 2 \leq x \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 3 } { 4 }\). Find
2. \(\mathrm { E } ( X )\),
3. the cumulative distribution function \(\mathrm { F } ( x )\).
4. Show that the median value of \(X\) lies between 2.70 and 2.75.

7. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0 \\ 2 x ^ { 2 } - x ^ { 3 } , & 0 \leqslant x \leqslant 1 \\ 1 , & x > 1 \end{cases}$$

1. Find \(\mathrm { P } ( X > 0.3 )\).
2. Verify that the median value of \(X\) lies between \(x = 0.59\) and \(x = 0.60\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Evaluate \(\mathrm { E } ( X )\).
5. Find the mode of \(X\).
6. Comment on the skewness of \(X\). Justify your answer.

1. The continuous random variable \(Y\) has cumulative distribution function \(\mathrm { F } ( y )\) given by

$$\mathrm { F } ( y ) = \left\{ \begin{array} { c l } 0 & y < 1 \\ k \left( y ^ { 4 } + y ^ { 2 } - 2 \right) & 1 \leqslant y \leqslant 2 \\ 1 & y > 2 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 18 }\).
2. Find \(\mathrm { P } ( Y > 1.5 )\).
3. Specify fully the probability density function f(y).

2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$

1. Find \(\mathrm { P } ( X < 0 )\).
2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
3. Write down the name of the distribution of \(X\).
4. Find the mean and the variance of \(X\).
5. Write down the value of \(\mathrm { P } ( X = 1 )\).

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

$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3 \\ 3 k & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\)
2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
3. Find the mean of \(X\).
4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)

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.

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.

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.

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. \(\mathrm { P } ( 2.3 < X < 3.4 )\).

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 $$\mathrm { P } ( \mu - k \sigma < X < \mu + k \sigma ) = 0.5$$

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 \(\mathrm { P } ( Y > 1 )\).

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

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 \(k\).

6. The continuous random variable \(X\) has a probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } k ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ k & 3 < x < 5 \\ k ( 6 - x ) & 5 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that the value of \(k\) is \(\frac { 1 } { 3 }\)
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Hence find the 90th percentile of the distribution.
5. Find \(\mathrm { P } [ \mathrm { E } ( X ) < X < 5.5 ]\)

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1. The length of time, \(T\), minutes, spent completing a particular task has probability density function

$$f ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } ( t - 1 ) & 1 < t \leqslant 2 \\ \frac { 1 } { 16 } \left( 14 t - 3 t ^ { 2 } - 8 \right) & 2 < t \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to find \(\mathrm { E } ( T )\) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 267 } { 40 }\)
2. find \(\operatorname { Var } ( T )\)
3. Find the cumulative distribution function \(\mathrm { F } ( t )\)
4. Find the 20th percentile of the time taken to complete the task.
5. Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,
6. find the probability that this person takes more than 3 minutes to complete the task.

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 1 \\ \frac { 4 } { 15 } ( x - 1 ) & 1 < x \leqslant 2 \\ k \left( \frac { a x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } \right) + b & 2 < x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$ where \(k , a\) and \(b\) are constants.
Given that the mode of \(X\) is \(\frac { 8 } { 3 }\)

1. show that \(a = 4\)
2. Find \(\mathrm { P } ( X < 2.5 )\) giving your answer to 3 significant figures.

5. The continuous random variable \(T\) represents the time in hours that students spend on homework. The cumulative distribution function of \(T\) is $$\mathrm { F } ( t ) = \begin{cases} 0 , & t < 0 \\ k \left( 2 t ^ { 3 } - t ^ { 4 } \right) & 0 \leq t \leq 1.5 \\ 1 , & t > 1.5 \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 16 } { 27 }\).
2. Find the proportion of students who spend more than 1 hour on homework.
3. Find the probability density function \(\mathrm { f } ( t )\) of \(T\).
4. Show that \(\mathrm { E } ( T ) = 0.9\).
5. Show that \(\mathrm { F } ( \mathrm { E } ( T ) ) = 0.4752\). A student is selected at random. Given that the student spent more than the mean amount of time on homework,
6. find the probability that this student spent more than 1 hour on homework.

7 Engineering work on the railway network causes an increase in the journey time of commuters travelling into work each morning. The increase in journey time, \(T\) hours, is modelled by a continuous random variable with probability density function $$\mathrm { f } ( t ) = \begin{cases} 4 t \left( 1 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { E } ( T ) = \frac { 8 } { 15 }\).
2. 1. Find the cumulative distribution function, \(\mathrm { F } ( t )\), for \(0 \leqslant t \leqslant 1\).
   2. Hence, or otherwise, for a commuter selected at random, find $$\mathrm { P } ( \text { mean } < T < \text { median } )$$