5.03a Continuous random variables: pdf and cdf

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 cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 3 \\ \frac { 1 } { 6 } ( x - 3 ) ^ { 2 } & 3 \leqslant x < 4 \\ \frac { x } { 3 } - \frac { 7 } { 6 } & 4 \leqslant x < c \\ 1 - \frac { 1 } { 6 } ( d - x ) ^ { 2 } & c \leqslant x < 7 \\ 1 & x \geqslant 7 \end{array} \right.$$ where \(c\) and \(d\) are constants.

1. Show that \(c = 6\)
2. Find \(\mathrm { P } ( X > 3.5 )\)
3. Find \(\mathrm { P } ( X > 4.5 \mid 3.5 < X < 5.5 )\)

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

2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$

1. Find \(\mathrm { P } ( X > 4 )\)
2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
3. Find the probability density function of \(X\), specifying it for all values of \(x\)
4. Write down the value of \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\)
6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)

5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\ 3 k & 2 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)

1. find the value of \(k\) and the value of \(a\)
2. Write down the mode of \(X\)

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1. A rectangle has a perimeter of 20 cm . The length, \(X \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 1 cm and 7 cm .

Find the probability that the length of the longer side of the rectangle is more than 6 cm long.

4. The lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 4 } { 9 } \left( x ^ { 2 } + 2 x - 3 \right) & 1 \leqslant x \leqslant 1.5 \\ 1 & x > 1.5 \end{array} \right.$$

1. Find the median of \(X\), giving your answer to 3 significant figures.
2. Find, in full, the probability density function of the random variable \(X\).
3. Find \(\mathrm { P } ( X \geqslant 1.2 )\) A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.
4. Find the probability that the lantern will still be working after 12 hours.

1. The random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by

$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k y ( a - y ) & 0 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are positive constants.

1. 1. Explain why \(a \geqslant 3\)
   2. Show that \(k = \frac { 2 } { 9 ( a - 2 ) }\) Given that \(\mathrm { E } ( Y ) = 1.75\)
2. show that \(a = 4\) and write down the value of \(k\). For these values of \(a\) and \(k\),
3. sketch the probability density function,
4. write down the mode of \(Y\).
   

4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable \(X\) represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of \(X\) and specify it fully.
2. Calculate the mean time of arrival of the bus.
3. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
4. Find the probability that Jean is late for work.

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.

An engineer measures, to the nearest cm , the lengths of metal rods.

1. Suggest a suitable model to represent the difference between the true lengths and the measured lengths.
2. Find the probability that for a randomly chosen rod the measured length will be within 0.2 cm of the true length. Two rods are chosen at random.
3. Find the probability that for both rods the measured lengths will be within 0.2 cm of their true lengths.
   

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 \(X\).

3. A rod of length \(2 l\) was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable \(X\).

1. Write down the name of the probability density function of \(X\), and specify it fully.
2. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
3. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
4. Find the probability that both of the shorter pieces are of length less than \(\frac { 1 } { 3 } l\).

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.

3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( \mathrm { X } )\),
4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).

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.

5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).

1. Write down the probability density function of \(X\), for all \(x\).
2. Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\). A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the standard deviation of \(X\),
5. the probability that the shorter piece of wire is at most 30 cm long.

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).

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

$$f ( x ) = \left\{ \begin{array} { c c } 2 ( x - 2 ) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the median of \(X\).
5. Comment on the skewness of this distribution. Give a reason for your answer.

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 \(x = 2.7\)