5.03a Continuous random variables: pdf and cdf

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 \(X\).

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( \sqrt { X } )\).
   The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the 40th percentile of \(Y\).

6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 28 } \left( e ^ { \frac { 1 } { 2 } x } + 4 e ^ { - \frac { 1 } { 2 } x } \right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = e ^ { \frac { 1 } { 2 } ( X ) }\).
2. Find the probability density function of \(Y\).
3. Find the 30th percentile of \(Y\).
4. Find \(\mathrm { E } \left( Y ^ { 4 } \right)\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

1 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } \left( x ^ { - \frac { 1 } { 3 } } - x ^ { - \frac { 2 } { 3 } } \right) & 1 \leqslant x \leqslant 27 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = X ^ { \frac { 1 } { 3 } }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).

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.

5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { ( x - 2 ) ^ { 2 } } { 12 } & 2 \leqslant x < 4 , \\ 1 - \frac { ( 8 - x ) ^ { 2 } } { 24 } & 4 \leqslant x \leqslant 8 , \\ 1 & x > 8 . \end{cases}$$ (c) Find the exact value of the interquartile range of \(X\).

6 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 60 } \left( 16 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$

1. Find the interquartile range of \(X\).
2. Find \(\mathrm { E } \left( X ^ { 3 } \right)\).
   The random variable \(Y\) is such that \(Y = \sqrt { X }\).
3. Find the probability density function of \(Y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2 \\ \frac { 1 } { 60 } x ^ { 2 } - \frac { 1 } { 15 } & 2 \leqslant x \leqslant 8 \\ 1 & x > 8 \end{cases}$$

1. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 6 )\).
2. Find \(\mathrm { E } ( \sqrt { X } )\).
3. Find \(\operatorname { Var } ( \sqrt { X } )\).
4. The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find the probability density function of \(Y\).

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 \(\operatorname { Var } ( X )\).

3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.

1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
2. State any assumption that you have made in part (a). \includegraphics[max width=\textwidth, alt={}, center]{44829994-2ef0-488d-aa3b-99fb0e36d733-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.

4 \includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463} The diagram shows the continuous random variable \(X\) with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4 \\ c & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .

1. Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
2. Find the exact value of the median of \(X\).
3. Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.

3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.

1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
2. State any assumption that you have made in part (a). \includegraphics[max width=\textwidth, alt={}, center]{b6635fbc-3c9d-4f93-b51a-b1cbd71ddbb1-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2 \\ \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.