Symmetry property of PDF

3 \includegraphics[max width=\textwidth, alt={}, center]{823cc2e5-e408-4b81-ac4d-7e9f584107cc-06_558_1077_260_523} The diagram shows the graph of the probability density function, f, of a random variable \(X\) that takes values between \(x = 0\) and \(x = 3\) only. The graph is symmetrical about the line \(x = 1.5\).

1. It is given that \(\mathrm { P } ( X < 0.6 ) = a\) and \(\mathrm { P } ( 0.6 < X < 1.2 ) = b\). Find \(\mathrm { P } ( 0.6 < X < 1.8 )\) in terms of \(a\) and \(b\).
2. It is now given that the equation of the probability density function of \(X\) is $$f ( x ) = \begin{cases} k x ^ { 2 } ( 3 - x ) ^ { 2 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
   1. Show that \(k = \frac { 10 } { 81 }\).
   2. Find \(\operatorname { Var } ( X )\).

6 The length of time, \(T\) minutes, that a passenger has to wait for a bus at a certain bus stop is modelled by the probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { 4000 } \left( 20 t - t ^ { 2 } \right) & 0 \leqslant t \leqslant 20 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(y = \mathrm { f } ( t )\).
2. Hence explain, without calculation, why \(\mathrm { E } ( T ) = 10\).
3. Find \(\operatorname { Var } ( T )\).
4. It is given that \(\mathrm { P } ( T < 10 + a ) = p\), where \(0 < a < 10\). Find \(\mathrm { P } ( 10 - a < T < 10 + a )\) in terms of \(p\).
5. Find \(\mathrm { P } ( 8 < T < 12 )\).
6. Give one reason why this model may be unrealistic.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The random variables \(X\) and \(W\) have probability density functions f and g defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = \begin{cases} p \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \\ \mathrm { g } ( w ) = \begin{cases} q \left( a ^ { 2 } - w ^ { 2 } \right) & - a \leqslant w \leqslant a \\ 0 & \text { otherwise } \end{cases} \end{gathered}$$ where \(a , p\) and \(q\) are constants.

1. 1. Write down the value of \(\mathrm { P } ( X \geqslant 0 )\).
   2. Write down the value of \(\mathrm { P } ( W \geqslant 0 )\).
   3. Write down an expression for \(q\) in terms of \(p\) only.
2. Given that \(\mathrm { E } ( X ) = 3\), find the value of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable \(X\) with probability density function $$f ( x ) = \begin{cases} - k ( x - 1 ) ( x - 3 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Without calculation, explain why \(\mathrm { E } ( X ) = 2\).
2. Show that \(k = \frac { 3 } { 4 }\).
3. Find \(\operatorname { Var } ( X )\).
   One turn consists of rolling the ball 3 times and noting the largest value of \(X\) obtained. If this largest value is greater than 2.5, the player scores a point.
4. Find the probability that on a particular turn the player scores a point.

7

1. \includegraphics[max width=\textwidth, alt={}, center]{1060d9f5-cf40-419e-b212-7266885c6617-3_465_1127_954_550} The diagram shows the graph of the probability density function of a variable \(X\). Given that the graph is symmetrical about the line \(x = 1\) and that \(\mathrm { P } ( 0 < X < 2 ) = 0.6\), find \(\mathrm { P } ( X > 0 )\).
2. A flower seller wishes to model the length of time that tulips last when placed in a jug of water. She proposes a model using the random variable \(X\) (in hundreds of hours) with probability density function given by $$f ( x ) = \begin{cases} k \left( 2.25 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1.5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
   1. Show that \(k = \frac { 4 } { 9 }\).
   2. Use this model to find the mean number of hours that a tulip lasts in a jug of water. The flower seller wishes to create a similar model for daffodils. She places a large number of daffodils in jugs of water and the longest time that any daffodil lasts is found to be 290 hours.
   3. Give a reason why \(\mathrm { f } ( x )\) would not be a suitable model for daffodils.
   4. The flower seller considers a model for daffodils of the form $$g ( x ) = \begin{cases} c \left( a ^ { 2 } - x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(c\) are constants. State a suitable value for \(a\). (There is no need to evaluate \(c\).)

6 \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable \(X\) that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line \(x = 1\). Between \(x = - 1\) and \(x = 3\) the graph is a quadratic curve. The random variable \(S\) is such that \(\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )\) and \(\operatorname { Var } ( S ) = \operatorname { Var } ( X )\).

1. On the grid below, sketch a quadratic graph for the probability density function of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_490_1191_1169_479} The random variable \(T\) is such that \(\mathrm { E } ( T ) = \mathrm { E } ( X )\) and \(\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )\).
2. On the grid below, sketch a quadratic graph for the probability density function of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{f45a7c6f-ebb4-43a2-8751-11aab3561d3c-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
3. Given that \(\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5\), show that \(a ^ { 3 } - 12 a + 8 = 0\).
4. Hence verify that \(0.69 < a < 0.70\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1195_255_475} The diagram shows the graph of the probability density function of a random variable \(X\) that takes values between - 1 and 3 only. It is given that the graph is symmetrical about the line \(x = 1\). Between \(x = - 1\) and \(x = 3\) the graph is a quadratic curve. The random variable \(S\) is such that \(\mathrm { E } ( S ) = 2 \times \mathrm { E } ( X )\) and \(\operatorname { Var } ( S ) = \operatorname { Var } ( X )\).

1. On the grid below, sketch a quadratic graph for the probability density function of \(S\). \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_490_1191_1169_479} The random variable \(T\) is such that \(\mathrm { E } ( T ) = \mathrm { E } ( X )\) and \(\operatorname { Var } ( T ) = \frac { 1 } { 4 } \operatorname { Var } ( X )\).
2. On the grid below, sketch a quadratic graph for the probability density function of \(T\). \includegraphics[max width=\textwidth, alt={}, center]{8f9e5f25-05c1-4a4e-9094-2a1b42416588-08_488_1187_1996_479} It is now given that $$f ( x ) = \begin{cases} \frac { 3 } { 32 } \left( 3 + 2 x - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
3. Given that \(\mathrm { P } ( 1 - a < X < 1 + a ) = 0.5\), show that \(a ^ { 3 } - 12 a + 8 = 0\).
4. Hence verify that \(0.69 < a < 0.70\).

6 A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that \(\mathrm { P } ( a < X < b ) = p\) and \(\mathrm { P } ( b < X < 3 ) = \frac { 13 } { 10 } p\), where \(p\) is a positive constant.

1. Show that \(p \leqslant \frac { 5 } { 23 }\).
2. Find \(\mathrm { P } ( b < X < 6 - a )\) in terms of \(p\).
   It is now given that the probability density function of \(X\) is f , where $$f ( x ) = \begin{cases} \frac { 1 } { 36 } \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$
3. Given that \(b = 2\) and \(p = \frac { 5 } { 27 }\), find the value of \(a\).

5 \includegraphics[max width=\textwidth, alt={}, center]{43403c12-93e6-44e4-b15e-e3c4363be5f9-08_254_634_260_717} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).

6 The distance travelled, in kilometres, by a Grippo brake pad before it needs to be replaced is modelled by \(10000 X\), where \(X\) is a random variable having the probability density function $$f ( x ) = \begin{cases} - k \left( x ^ { 2 } - 5 x + 6 \right) & 2 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ The graph of \(y = \mathrm { f } ( x )\) is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c1dcf0f5-e971-4afd-81ca-4d860732825c-3_439_1100_580_520}

1. Show that \(k = 6\).
2. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Sami fits four new Grippo brake pads on his car. Find the probability that at least one of these brake pads will need to be replaced after travelling less than 22000 km .

7

1. \includegraphics[max width=\textwidth, alt={}, center]{3f1a0c67-03a4-4b4f-99c0-4336ba7d56b0-3_255_643_264_790} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
   2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
   3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).
2. A random variable, \(W\), has probability density function given by $$\mathrm { g } ( w ) = \begin{cases} a w & 0 \leqslant w \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. Given that the median of \(W\) is 2 , find \(a\) and \(b\).

6 The graph of the probability density function f of a random variable \(X\) is symmetrical about the line \(x = 2\). It is given that \(\mathrm { P } ( 2 < X < 5 ) = \frac { 117 } { 256 }\).

1. Using only this information show that \(\mathrm { P } ( X > - 1 ) = \frac { 245 } { 256 }\).
   It is now given that, for \(x\) in a suitable domain, $$f ( x ) = k \left( 12 + 4 x - x ^ { 2 } \right) , \text { where } k \text { is a constant. }$$
2. Find the value of \(k\).
3. A different random variable \(X\) has probability density function \(\mathbf { g } ( x ) = \frac { 2 } { 9 } \left( 2 + x - x ^ { 2 } \right)\). The domain of \(X\) is all values of \(x\) for which \(\mathrm { g } ( x ) \geqslant 0\). Find \(\operatorname { Var } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{ff3433b0-baab-45e3-845e-56a794739bba-11_63_1547_447_347}

5 Bottles of Lanta contain approximately 300 ml of juice. The volume of juice, in millilitres, in a bottle is \(300 + X\), where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4000 } \left( 100 - x ^ { 2 } \right) & - 10 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the probability that a randomly chosen bottle of Lanta contains more than 305 ml of juice.
2. Given that \(25 \%\) of bottles of Lanta contain more than \(( 300 + p ) \mathrm { ml }\) of juice, show that $$p ^ { 3 } - 300 p + 1000 = 0$$
3. Given that \(p = 3.47\), and that \(50 \%\) of bottles of Lanta contain between ( \(300 - q\) ) and ( \(300 + q\) ) ml of juice, find \(q\). Justify your answer.

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

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

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.

The graph of the probability density function of a random variable \(X\) is symmetrical about the line \(x = 4\). Given that \(\text{P}(X < 5) = \frac{20}{39}\), find \(\text{P}(3 < X < 5)\). [2]

\includegraphics{figure_7} The diagram shows the graph of the probability density function, f, of a random variable \(X\) which takes values between \(-3\) and 2 only.

1. Given that the graph is symmetrical about the line \(x = -0.5\) and that P(\(X < 0\)) = \(p\), find P(\(-1 < X < 0\)) in terms of \(p\). [2]
2. It is now given that the probability density function shown in the diagram is given by $$\text{f}(x) = \begin{cases} a - b(x^2 + x) & -3 \leq x \leq 2, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are positive constants.
   1. Show that \(30a - 55b = 6\). [3]
   2. By substituting a suitable value of \(x\) into f(\(x\)), find another equation relating \(a\) and \(b\) and hence determine the values of \(a\) and \(b\). [3]

A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that P\((a < X < b) = p\) and P\((b < X < 3) = \frac{13}{10}p\), where \(p\) is a positive constant.

1. Show that \(p \leq \frac{5}{23}\). [1]
2. Find P\((b < X < 6 - a)\) in terms of \(p\). [2]

It is now given that the probability density function of \(X\) is \(f\), where $$f(x) = \begin{cases} \frac{1}{36}(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise}. \end{cases}$$

1. Given that \(b = 2\) and \(p = \frac{5}{81}\), find the value of \(a\). [5]