5.03c Calculate mean/variance: by integration

5 Participants in a school jumping competition gain a total score for each jump based on the length, \(L\) metres, jumped beyond a fixed point and a mark, \(S\), for style. \(L\) may be regarded as a continuous random variable with probability density function $$\mathrm { f } ( l ) = \left\{ \begin{array} { c c } w l & 0 \leq l \leq 15 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(w\) is a constant. \(S\) may be regarded as a discrete random variable with probability function $$\mathrm { P } ( S = s ) = \left\{ \begin{array} { c l } \frac { 1 } { 15 } s & s = 1,2,3,4,5 \\ 0 & \text { otherwise } \end{array} \right.$$ Assume that \(L\) and \(S\) are independent. The total score for a participant in this competition, \(T\), is given by \(T = L ^ { 2 } + \frac { 1 } { 2 } S\) Show that the expected total score for a participant is \(114 \frac { 1 } { 3 }\)

6 The continuous random variable \(T\) has probability density function defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 3 } & 0 \leq t \leq \frac { 3 } { 2 } \\ \frac { 9 - 2 t } { 18 } & \frac { 3 } { 2 } \leq t \leq \frac { 9 } { 2 } \\ 0 & \text { otherwise } \end{array} \right.$$ 6

1. 1. Sketch this probability density function below. \includegraphics[max width=\textwidth, alt={}, center]{6ccf7d1d-5a7b-47d1-b38e-c7e762204746-07_1009_1041_1073_520} 6
      1. (ii) State the median of \(T\). 6
   2. 1. Find \(\mathrm { E } ( T )\) [0pt] [2 marks]
         6
   3. (ii) Given that \(\mathrm { E } \left( T ^ { 2 } \right) = \frac { 15 } { 4 }\), find \(\operatorname { Var } ( 4 T - 5 )\) [3 marks]

8 A random sample of 100 students were given a task and the time taken by each student to complete the task was recorded. The maximum time allowed to complete the task was one minute and all students completed the task within the maximum time. The times, \(T\) minutes, for the random sample of students are summarised as follows. \(n = 100 \quad \sum t = 61.88\) A researcher proposes that \(T\) can be modelled by the continuous random variable with probability density function \(f ( t ) = \begin{cases} \alpha t ^ { \alpha - 1 } & 0 \leqslant t \leqslant 1 , \\ 0 & \text { otherwise, } \end{cases}\) where \(\alpha\) is a positive constant. \section*{(a) In this question you must show detailed reasoning.} By finding \(\mathbf { E } ( T )\) according to the researcher's model, determine an approximation for the value of \(\alpha\). Give your answer correct to \(\mathbf { 3 }\) significant figures. Further information about the times taken for the sample of 100 students to complete the task is given in the table. (b) Using the value of \(\alpha\) found in part (a), determine the extent to which the proposed model is a good model. (Do not carry out a goodness of fit test.)

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 \(Y ^ { 2 }\).

8 A continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } 0.8 \mathrm { e } ^ { - 0.8 x } & x \geq 0 \\ 0 & x < 0 \end{array} \right.$$

1. Find the mean and variance of \(X\). The lifetime of a certain organism is thought to have the same distribution as \(X\). The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.

   Range	\(0 \leq x < 1\)	\(1 \leq x < 2\)	\(2 \leq x < 3\)	\(3 \leq x < 4\)	\(x \geq 4\)
   Observed	24	22	10	3	1
   Expected	33.040	14.846	6.671	2.997	2.446

2. Show how the expected frequency for \(1 \leq x < 2\) is obtained.
3. Carry out a goodness of fit test at the \(5 \%\) significance level.

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

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

5. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( x ^ { 2 } + a \right) & - 1 < x \leqslant 2 \\ 3 k & 2 < x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
Given that \(\mathrm { E } ( X ) = \frac { 17 } { 12 }\)

1. find the value of \(k\) and the value of \(a\)
2. Write down the mode of \(X\)

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)

1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
2. find the value of \(\alpha\) and the value of \(\beta\)
3. find \(\operatorname { Var } ( X )\)
4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)

1. The time, in thousands of hours, that a certain electrical component will last is modelled by the random variable \(X\), with probability density function

$$f ( x ) = \begin{cases} \frac { 3 } { 64 } x ^ { 2 } ( 4 - x ) & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ Using this model, find, by algebraic integration,

1. the mean number of hours that a component will last,
2. the standard deviation of \(X\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ce1f9aa7-cf16-4293-98b1-157eed35b761-06_478_974_1069_479} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of the probability density function of the random variable \(X\).
3. Give a reason why the random variable \(X\) might be unsuitable as a model for the time, in thousands of hours, that these electrical components will last.
4. Sketch a probability density function of a more realistic model.

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. Albert uses scales in his kitchen to weigh some fruit.

The random variable \(D\) represents, in grams, the weight of the fruit given by the scales minus the true weight of the fruit. The random variable \(D\) is uniformly distributed over the interval \([ - 2.5,2.5 ]\)

1. Specify the probability density function of \(D\)
2. Find the standard deviation of \(D\) Albert weighs a banana on the scales.
3. Write down the probability that the weight given by the scales equals the true weight of the banana.
4. Find the probability that the weight given by the scales is within 1 gram of the banana's true weight. Albert weighs 10 bananas on the scales, one at a time.
5. Find the probability that the weight given by the scales is within 1 gram of the true weight for at least 6 of the bananas.

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.

1. A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that \(5 \%\) of people who buy a ticket do not turn up for the journey.

The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable \(X\) represents the number of these people who do not turn up for the journey.

1. State one assumption required to model \(X\) as a binomial distribution. For this week's journey find,
2. 1. the probability that all 50 people turn up for the journey,
   2. \(\mathrm { P } ( X = 1 )\) The bus company receives \(\pounds 20\) for each ticket sold and all 50 tickets are sold. It must pay out \(\pounds 60\) to each person who buys a ticket and turns up for the journey but does not have a seat.
3. Find the bus company's expected total earnings per journey.
   

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 \(X\) is uniformly distributed over the interval \([ a , b ]\) where \(0 < a < b\)

Given that \(\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }\)

1. 1. show that \(\mathrm { E } ( X ) = \frac { 5 a } { 2 }\)
   2. find \(\mathrm { P } ( X > b - 4 a )\) The continuous random variable \(Y\) is uniformly distributed over the interval [3, c] where \(c > 3\) Given that \(\operatorname { Var } ( Y ) = 3 c - 9\), find
2. 1. the value of \(c\)
   2. \(\mathrm { P } ( 2 Y - 7 < 20 - Y )\)
   3. \(\mathrm { E } \left( Y ^ { 2 } \right)\)

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}
   

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5. A piece of wood \(A B\) is 3 metres long. The wood is cut at random at a point \(C\) and the random variable \(W\) represents the length of the piece of wood \(A C\).

1. Find the probability that the length of the piece of wood \(A C\) is more than 1.8 metres. The two pieces of wood \(A C\) and \(C B\) form the two shortest sides of a right-angled triangle. The random variable \(X\) represents the length of the longest side of the right-angled triangle.
2. Show that \(X ^ { 2 } = 2 W ^ { 2 } - 6 W + 9\) [0pt] [You may assume for random variables \(S , T\) and \(U\) and for constants \(a\) and \(b\) that if \(S = a T + b U\) then \(\mathrm { E } ( S ) = a \mathrm { E } ( T ) + b \mathrm { E } ( U ) ]\)
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. Find \(\mathrm { P } \left( X ^ { 2 } > 5 \right)\)

2 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < - k \\ \frac { x + k } { 4 k } & - k \leqslant x \leqslant 3 k \\ 1 & x > 3 k \end{array} \right.$$ where \(k\) is a positive constant.

1. Specify fully, in terms of \(k\), the probability density function of \(X\)
2. Write down, in terms of \(k\), the value of \(\mathrm { E } ( X )\)
3. Show that \(\operatorname { Var } ( X ) = \frac { 4 } { 3 } k ^ { 2 }\)
4. Find, in terms of \(k\), the value of \(\mathrm { E } \left( 3 X ^ { 2 } \right)\)

4 The continuous random variable \(X\) has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3 \\ k & 3 < x \leqslant 6 \\ \frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\)
2. Show that \(k = \frac { 1 } { 6 }\)
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 61 } { 12 }\)
4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
5. Describe the skewness of the distribution, giving a reason for your answer.
   

7 The sides of a square are each of length \(L \mathrm {~cm}\) and its area is \(A \mathrm {~cm} ^ { 2 }\) Given that \(A\) is uniformly distributed on the interval [10,30]

1. find \(\mathrm { P } ( L \geqslant 4.5 )\)
2. find \(\operatorname { Var } ( L )\)
   \includegraphics[max width=\textwidth, alt={}]{a009b02e-4cd3-497b-a141-4630c653e20b-28_2655_1947_114_116}

The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\), shown in the diagram, where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{f4fa6add-5860-4c88-bb70-f3edd9b22211-12_511_1096_351_351}

1. Find \(\mathrm { P } ( X < 10 k )\)
2. Show that \(k = \frac { 1 } { \pi }\)
3. Find, in terms of \(\pi\), the values of
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\) Circles are drawn with area \(A\), where $$A = \pi \left( X + \frac { 2 } { \pi } \right) ^ { 2 }$$
   (d) Find \(\mathrm { E } ( A )\)

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 0 \\ a x + b x ^ { 2 } & 0 \leqslant x \leqslant k \\ 1 & x > k \end{array} \right.$$ where \(a , b\) and \(k\) are positive constants.

1. Show that \(a k = 1 - b k ^ { 2 }\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\)
2. show that \(5 b k ^ { 3 } = 36 - 15 k\) Using part (a) and given that \(\mathrm { E } ( X ) = \frac { 6 } { 5 }\) and \(\operatorname { Var } ( X ) = \frac { 22 } { 75 }\)
3. show that \(5 b k ^ { 4 } = 52 - 10 k ^ { 2 }\) Given that \(k < 3\)
4. find the value of \(k\)
5. Hence find the value of \(a\) and the value of \(b\)