5.03f Relate pdf-cdf: medians and percentiles

6 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.4 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( X > 2 )\).
2. Find the interquartile range of \(X\).

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 x ^ { 2 } } + \frac { 1 } { 4 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find the exact value of the interquartile range of \(X\).

6 The random variable \(T\) is the lifetime, in hours, of a randomly chosen battery of a particular type. It is given that \(T\) has a negative exponential distribution with mean 400 hours.

1. Write down the probability density function of \(T\).
2. Find the probability that a battery of this type has a lifetime that is less than 500 hours.
3. Find the median of the distribution.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\). Show that \(Y\) has probability density function given by $$\operatorname { g } ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16 \\ 0 & \text { otherwise } \end{cases}$$ Find

1. the median value of \(Y\),
2. the expected value of \(Y\).

7 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Find the median of \(X\).

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } x ^ { 2 } & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is defined by \(Y = X ^ { 2 }\).

1. Show that \(Y\) has probability density function given by $$g ( y ) = \begin{cases} \frac { 1 } { 42 } y ^ { \frac { 1 } { 2 } } & 1 \leqslant y \leqslant 16 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the median value of \(Y\).
3. Find the expected value of \(Y\).

7 The continuous random variable \(X\) has probability density function \(f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 , \\ 0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant and \(n\) is a parameter whose value is positive. It is given that the median of \(X\) is 0.8816 correct to 4 decimal places. Ten independent observations of \(X\) are obtained. Find the expected number of observations that are less than 0.8 .

8 The continuous random variable \(Y\) has a uniform distribution on [0,2].

1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 6 } x ( x + 1 ) & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$

1. Find the value of \(a\) such that \(\mathrm { P } ( X > a ) = 0.4\) Give your answer to 3 significant figures.
2. Use calculus to find (i) \(\mathrm { E } ( X )\) (ii) \(\operatorname { Var } ( X )\).

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

$$f ( x ) = \left\{ \begin{array} { c c } k ( x + 1 ) ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ k ( 6 - 2 x ) & 1 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\).
2. Show that the value of \(k\) is \(\frac { 3 } { 20 }\)
3. Define fully the cumulative distribution function \(\mathrm { F } ( x )\).
4. Find the median of \(X\), giving your answer to 3 significant figures.

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 \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)

7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \begin{cases} \frac { 1 } { 20 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 10 } ( 6 - x ) & 2 < x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\).
3. Show that \(\mathrm { P } ( X > 2 ) = 0.8\)
4. Define fully the cumulative distribution function \(\mathrm { F } ( x )\). Given that \(\mathrm { P } ( X < a \mid X > 2 ) = \frac { 5 } { 8 }\)
5. find the value of \(\mathrm { F } ( a )\).
6. Hence, or otherwise, find the value of \(a\). Give your answer to 3 significant figures.

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 16 } ( x - 1 ) ^ { 2 } & 1 \leqslant x \leqslant 5 \\ 1 & x > 5 \end{array} \right.$$

1. Find \(\mathrm { P } ( X > 4 )\)
2. Find \(\mathrm { P } ( X > 3 \mid 2 < X < 4 )\)
3. Find the exact value of \(\mathrm { E } ( X )\)

1. In a local council, \(60 \%\) of households recycle at least half of their waste. A random sample of 80 households is taken.

The random variable \(X\) represents the number of households in the sample that recycle at least half of their waste.

1. Using a suitable approximation, find the smallest number of households, \(n\), such that $$\mathrm { P } ( X \geqslant n ) < 0.05$$ The number of bags recycled per family per week was known to follow a Poisson distribution with mean 1.5 Following a recycling campaign, the council believes the mean number of bags recycled per family per week has increased. To test this belief, 6 families are selected at random and the total number of bags they recycle the following week is recorded. The council wishes to test, at the 5\% level of significance, whether or not there is evidence that the mean number of bags recycled per family per week has increased.
2. Find the critical region for the total number of bags recycled by the 6 families.

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 \(X\) 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 \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)

   Leave blank	Q7

2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable \(W\) with cumulative distribution function $$\mathrm { F } ( w ) = \left\{ \begin{array} { c c } 0 & w < 0 \\ \frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4 \\ 1 & w > 4 \end{array} \right.$$

1. Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. A random sample of 30 novice tightrope artists is taken.
2. Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. Given \(\mathrm { E } ( W ) = 1.6\)
3. use algebraic integration to find \(\operatorname { Var } ( W )\) DO NOT WRITEIN THIS AREA

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}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

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.

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.