5.03b Solve problems: using pdf

5 The length, \(X \mathrm {~cm}\), of a piece of wooden planking is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(b\) is a positive constant.

1. Find the mean and variance of \(X\) in terms of \(b\). The lengths of a random sample of 100 pieces were measured and it was found that \(\Sigma x = 950\).
2. Show that the value of \(b\) estimated from this information is 19 . Using this value of \(b\),
3. find the probability that the length of a randomly chosen piece is greater than 11 cm ,
4. find the probability that the mean length of a random sample of 336 pieces is less than 9 cm .

7 The time in hours taken for clothes to dry can be modelled by the continuous random variable with probability density function given by $$f ( t ) = \begin{cases} k \sqrt { } t & 1 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 14 }\).
2. Find the mean time taken for clothes to dry.
3. Find the median time taken for clothes to dry.
4. Find the probability that the time taken for clothes to dry is between the mean time and the median time.

5 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \cos x & 0 \leqslant x \leqslant \frac { 1 } { 4 } \pi \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \sqrt { } 2\).
2. Find \(\mathrm { P } ( X > 0.4 )\).
3. Find the upper quartile of \(X\).
4. Find the probability that exactly 3 out of 5 random observations of \(X\) have values greater than the upper quartile.

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

1. Show that the value of \(k\) is \(\frac { 32 } { 9 }\).
2. Find \(\mathrm { E } ( X )\).
3. Is the median less than or greater than 1.5? Justify your answer numerically.

4 \includegraphics[max width=\textwidth, alt={}, center]{c7cbd61b-9a62-494a-b595-f624ec5c0bea-2_351_561_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).

4 \includegraphics[max width=\textwidth, alt={}, center]{0784d885-5710-4eb4-8cf8-2582122bf7ed-2_351_554_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).

5 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the median of \(X\).
3. Two independent values of \(X\) are chosen at random. Find the probability that both these values are greater than 3 .

6 Darts are thrown at random at a circular board. The darts hit the board at distances \(X\) centimetres from the centre, where \(X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { a ^ { 2 } } x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant.

1. Verify that f is a probability density function whatever the value of \(a\). It is now given that \(\mathrm { E } ( X ) = 8\).
2. Find the value of \(a\).
3. Find the probability that a dart lands more than 6 cm from the centre of the board.

5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 8 }\).
2. 20\% of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
3. Find \(\mathrm { E } ( X )\).

5 The volume, in \(\mathrm { cm } ^ { 3 }\), of liquid left in a glass by people when they have finished drinking all they want is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 8 }\).
2. \(20 \%\) of people leave at least \(d \mathrm {~cm} ^ { 3 }\) of liquid in a glass. Find \(d\).
3. Find \(\mathrm { E } ( X )\).

3 The waiting time, \(T\) weeks, for a particular operation at a hospital has probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2500 } \left( 100 t - t ^ { 3 } \right) & 0 \leqslant t \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. Given that \(\mathrm { E } ( T ) = \frac { 16 } { 3 }\), find \(\operatorname { Var } ( T )\).
2. \(10 \%\) of patients have to wait more than \(n\) weeks for their operation. Find the value of \(n\), giving your answer correct to the nearest integer.

3

1. The time for which Lucy has to wait at a certain traffic light each day is \(T\) minutes, where \(T\) has probability density function given by $$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
2. \includegraphics[max width=\textwidth, alt={}, center]{f9436a68-ec88-4feb-9c06-fc29fe53d1fe-2_405_793_1471_715} The diagram shows the graph of the probability density function, g , of a random variable \(X\). The graph of g is a semicircle with centre \(( 0,0 )\) and radius \(a\). Elsewhere \(\mathrm { g } ( x ) = 0\).
   1. Find the value of \(a\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Given that \(\mathrm { P } ( X < - c ) = 0.2\), find \(\mathrm { P } ( X < c )\).

3

1. The time for which Lucy has to wait at a certain traffic light each day is \(T\) minutes, where \(T\) has probability density function given by $$f ( t ) = \begin{cases} \frac { 3 } { 2 } t - \frac { 3 } { 4 } t ^ { 2 } & 0 \leqslant t \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ Find the probability that, on a randomly chosen day, Lucy has to wait for less than half a minute at the traffic light.
2. \includegraphics[max width=\textwidth, alt={}, center]{c08d3228-430e-4158-9362-1655deb1feb7-2_405_791_1471_715} The diagram shows the graph of the probability density function, g , of a random variable \(X\). The graph of g is a semicircle with centre \(( 0,0 )\) and radius \(a\). Elsewhere \(\mathrm { g } ( x ) = 0\).
   1. Find the value of \(a\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Given that \(\mathrm { P } ( X < - c ) = 0.2\), find \(\mathrm { P } ( X < c )\).

2 \includegraphics[max width=\textwidth, alt={}, center]{323cf83a-e23b-494e-a911-856d8f1c92fd-2_483_791_708_676} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(c\).
2. Find the value of \(a\) such that \(\mathrm { P } ( a < X < 1 ) = 0.1\).
3. Find \(\mathrm { E } ( X )\).

4 A random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 2 } { 3 }\).
2. Find the median of \(X\).

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

1. Show that \(k = \frac { 3 } { 32 }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\) and hence write down the value of \(\mathrm { E } ( X )\).
3. Find \(\mathrm { P } ( X < 1 )\).

6 \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_504_260_534} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_373_495_260_1123} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_371_497_776_534} \includegraphics[max width=\textwidth, alt={}, center]{e0ad3268-117e-4a0c-942d-84ee148d8907-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.

1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
   2. Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
   3. Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).

6 \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_504_260_534} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_373_495_260_1123} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_371_497_776_534} \includegraphics[max width=\textwidth, alt={}, center]{1e20bcc7-a501-4df0-9d49-cca2db4c279a-3_367_488_778_1128} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between 0 and 3 only, and their medians are \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) respectively.

1. List \(m _ { W } , m _ { X } , m _ { Y }\) and \(m _ { Z }\) in order of size, starting with the largest.
2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 4 } { 81 } x ^ { 3 } & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(\mathrm { E } ( X ) = \frac { 12 } { 5 }\).
   2. Calculate \(\mathrm { P } ( X > \mathrm { E } ( X ) )\).
   3. Write down the value of \(\mathrm { P } ( X < 2 \mathrm { E } ( X ) )\).

8 \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.

1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
   2. Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
   3. Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x < 2 \\ k ( 6 - x ) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 40 }\).
2. Given that \(\mathrm { E } ( X ) = 2.5\), find \(\operatorname { Var } ( X )\).
3. Find the median value of \(X\).

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

1. Find \(\operatorname { Var } ( \sqrt { X } )\).
   The continuous random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

2 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < - 1 \\ \frac { 1 } { 2 } ( 1 + x ) ^ { 2 } & - 1 \leqslant x \leqslant 0 \\ 1 - \frac { 1 } { 2 } ( 1 - x ) ^ { 2 } & 0 < x \leqslant 1 \\ 1 & x > 1 \end{array} \right.$$

1. Find the probability density function of \(X\).
2. Find \(\mathrm { P } \left( - \frac { 1 } { 2 } \leqslant X \leqslant \frac { 1 } { 2 } \right)\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 5 } x & 0 \leqslant x < 1 \\ 2 a - \frac { 1 } { 5 } x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Find the cumulative distribution function of \(X\).

5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ 1 - \frac { 1 } { 144 } ( 12 - x ) ^ { 2 } & 0 \leqslant x \leqslant 12 \\ 1 & x > 12 \end{cases}$$

1. Find the upper quartile of \(X\).
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).
   The random variable \(Y\) is given by \(Y = \sqrt { X }\).
3. Find the probability density function of \(Y\).

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

1. Show that \(k = \frac { 2 } { 5 }\).
2. Find the interquartile range of \(X\).
3. Find \(\operatorname { Var } ( X )\).