5.03a Continuous random variables: pdf and cdf

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 .

6 At a certain shop the weekly demand, in kilograms, for flour is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} k x ^ { - \frac { 1 } { 2 } } & 4 \leqslant x \leqslant 25 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 6 }\).
2. Calculate the mean weekly demand for flour at the shop.
3. At the beginning of one week, the shop has 20 kg of flour in stock. Find the probability that this will not be enough to meet the demand for that week.
4. Give a reason why the model may not be realistic.

7 \includegraphics[max width=\textwidth, alt={}, center]{7333c047-edad-4385-b3f8-248e8725cfcb-3_412_718_1037_715} A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \sin x & 0 \leqslant x \leqslant \frac { 2 } { 3 } \pi \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant, as shown in the diagram.

1. Show that \(k = \frac { 2 } { 3 }\).
2. Show that the median of \(X\) is 1.32 , correct to 3 significant figures.
3. Find \(\mathrm { E } ( X )\).

6 The time in minutes taken by people to read a certain booklet is modelled by the random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2 \sqrt { } t } & 4 \leqslant t \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the time within which \(90 \%\) of people finish reading the booklet.
2. Find \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).

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

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
3. Hence explain whether the mean of \(X\) is less than, equal to or greater than the median of \(X\).

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

1. Show that \(k = 2\).
2. Find \(\mathrm { P } ( 1 \leqslant X \leqslant 2 )\).
3. Find \(\mathrm { E } ( X )\).

7 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. Show that \(k = \frac { 1 } { \ln a }\).
2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
3. Find the median of \(X\) in terms of \(a\).

2 \includegraphics[max width=\textwidth, alt={}, center]{43b2498f-73e2-4d33-adaf-fc3e460fa36a-2_358_1093_495_520} A random variable \(X\) takes values between 0 and 4 only and has probability density function as shown in the diagram. Calculate the median of \(X\).

5 The lifetime, \(X\) years, of a certain type of battery has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. State what the value of \(a\) represents in this context.
2. Show that \(k = \frac { a } { a - 1 }\).
3. Experience has shown that the longest that any battery of this type lasts is 2.5 years. Find the mean lifetime of batteries of this type.

1 \includegraphics[max width=\textwidth, alt={}, center]{cfffe79d-91c9-48b8-a3e6-887d7891441d-2_478_691_260_724} The random variable \(X\) has probability density function, f , as shown in the diagram, where \(a\) is a constant. Find the value of \(a\) and hence show that \(\mathrm { E } ( X ) = 0.943\) correct to 3 significant figures. [5]

6 The waiting time, \(T\) minutes, for patients at a doctor's surgery has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} k \left( 225 - t ^ { 2 } \right) & 0 \leqslant t \leqslant 15 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2250 }\).
2. Find the probability that a patient has to wait for more than 10 minutes.
3. Find the mean waiting time.

6 The time, in minutes, taken by people to complete a test is modelled by the continuous random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 5 \leqslant x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 10\).
2. Show that \(\mathrm { E } ( X ) = 10 \ln 2\).
3. Find \(\mathrm { P } ( X > 9 )\).
4. Given that \(\mathrm { P } ( X < a ) = 0.6\), find \(a\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b054d0a0-01b6-4785-807c-851551b90544-06_382_743_260_699} The diagram shows the probability density function, f , of a random variable \(X\), in terms of the constants \(a\) and \(b\).

1. Find \(b\) in terms of \(a\).
2. Show that \(\mathrm { f } ( x ) = \frac { 2 } { a } - \frac { 2 } { a ^ { 2 } } x\).
3. Given that \(\mathrm { E } ( X ) = 0.5\), find \(a\).

4

1. 
   
   [diagram]
   The diagram shows the graph of the probability density function, f , of a random variable \(X\), where \(a\) is a constant greater than 0.5 . The graph between \(x = 0\) and \(x = a\) is a straight line parallel to the \(x\)-axis.
   1. Find \(\mathrm { P } ( X < 0.5 )\) in terms of \(a\).
   2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
   3. Show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } a ^ { 2 }\).
2. A random variable \(T\) has probability density function given by $$\operatorname { g } ( t ) = \begin{cases} \frac { 3 } { 2 ( t - 1 ) ^ { 2 } } & 2 \leqslant t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Find the value of \(b\) such that \(\mathrm { P } ( T \leqslant b ) = \frac { 3 } { 4 }\).

5 The manufacturer of a certain type of biscuit claims that \(10 \%\) of packets include a free offer printed on the packet. Jyothi suspects that the true proportion is less than \(10 \%\). He plans to test the claim by looking at 40 randomly selected packets and, if the number which include the offer is less than 2 , he will reject the manufacturer's claim.

1. State suitable hypotheses for the test.
2. Find the probability of a Type I error.
   On another occasion Jyothi looks at 80 randomly selected packets and finds that exactly 6 include the free offer.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of packets that include the offer.
4. Use your confidence interval to comment on the manufacturer's claim. \(6 X\) is a random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

6 A function f is defined by $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { a ^ { 3 } } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that f is a probability density function for all positive values of \(a\).
   The random variable \(X\) has probability density function f and the median of \(X\) is 2 .
2. Show that \(a = 2.52\), correct to 3 significant figures.
3. Find \(\mathrm { E } ( X )\).

6 The probability density function, f, of a random variable \(X\) is given by $$f ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{189bcf7b-279f-457b-8232-ace7f0c9797f-05_456_668_260_735} The random variable \(X\) takes values in the range \(1 \leqslant x \leqslant p\), where \(p\) is a constant. The graph of the probability density function of \(X\) is shown in the diagram.

1. Show that \(p = 2\).
2. Find \(\mathrm { E } ( X )\).

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 \(b\).

5

1. \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_292_517_264_338} \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_289_518_264_858} \includegraphics[max width=\textwidth, alt={}, center]{61ba010c-d6a2-4c19-9998-0ae048244a32-06_273_510_365_1377} The diagram shows the graphs of three functions, \(f _ { 1 } , f _ { 2 }\) and \(f _ { 3 }\). The function \(f _ { 1 }\) is a probability density function.
   1. State the value of \(k\).
   2. For each of the functions \(\mathrm { f } _ { 2 }\) and \(\mathrm { f } _ { 3 }\), state why it cannot be a probability density function.
2. The probability density function g is defined by $$g ( x ) = \begin{cases} 6 \left( a ^ { 2 } - x ^ { 2 } \right) & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
   1. Show that \(a = \frac { 1 } { 2 }\).
   2. State the value of \(\mathrm { E } ( X )\).
   3. Find \(\operatorname { Var } ( X )\).

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}

4 \includegraphics[max width=\textwidth, alt={}, center]{65b50bfb-5fd8-4cf3-ae3b-cffc12e23cd8-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ec7cab36-683b-4022-9cac-fb3b4e64778a-04_332_1100_260_520} A random variable \(X\) takes values between 0 and 3 only and has probability density function as shown in the diagram, where \(c\) is a constant.

1. Show that \(c = \frac { 2 } { 3 }\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Calculate \(\mathrm { E } ( X )\).

4 \includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-07_316_984_260_577} The diagram shows the probability density function, \(\mathrm { f } ( x )\), of a random variable \(X\). For \(0 \leqslant x \leqslant a\), \(\mathrm { f } ( x ) = k\); elsewhere \(\mathrm { f } ( x ) = 0\).

1. Express \(k\) in terms of \(a\).
2. Given that \(\operatorname { Var } ( X ) = 3\), find \(a\).

4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants. It is given that \(\mathrm { E } ( X ) = 3\).

1. Find the value of \(a\) and show that \(k = \frac { 1 } { 6 }\).
2. Find the median of \(X\).