5.03c Calculate mean/variance: by integration

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts. A supermarket receives complaints at a mean rate of 6 per week.

1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
   1. fewer than 3 complaints received by the supermarket,
   2. at least 6 complaints received by the supermarket. In a randomly selected week, the supermarket received 12 complaints.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
4. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
   1. The continuous random variable \(Y\) has cumulative distribution function given by
   $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
   1. The discrete random variable \(X\) is given by
   $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$

1. show that \(\sigma = 6\)
2. Hence find the value of \(n\) and the value of \(p\)

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 \(X\)

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1. The random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) given by

$$\mathrm { f } ( y ) = \left\{ \begin{array} { c c } k y ( a - y ) & 0 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are positive constants.

1. 1. Explain why \(a \geqslant 3\)
   2. Show that \(k = \frac { 2 } { 9 ( a - 2 ) }\) Given that \(\mathrm { E } ( Y ) = 1.75\)
2. show that \(a = 4\) and write down the value of \(k\). For these values of \(a\) and \(k\),
3. sketch the probability density function,
4. write down the mode of \(Y\).
   

4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable \(X\) represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of \(X\) and specify it fully.
2. Calculate the mean time of arrival of the bus.
3. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
4. Find the probability that Jean is late for work.

7. A continuous random variable \(X\) has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 0 \\ k x ^ { 2 } + 2 k x , & 0 \leq x \leq 2 \\ 8 k , & x > 2 \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 8 }\).
2. Find the median of \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
5. Write down the mode of \(X\).
6. Find \(\mathrm { E } ( X )\).
7. Comment on the skewness of this distribution.

4. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0 \\ \frac { 1 } { 3 } x ^ { 2 } \left( 4 - x ^ { 2 } \right) , & 0 \leq x \leq 1 \\ 1 & x > 1 \end{cases}$$

1. Find \(\mathrm { P } ( X > 0.7 )\).
2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
3. Calculate \(\mathrm { E } ( X )\) and show that, to 3 decimal places, \(\operatorname { Var } ( X ) = 0.057\). One measure of skewness is $$\frac { \text { Mean - Mode } } { \text { Standard deviation } } .$$
4. Evaluate the skewness of the distribution of \(X\).

3. A rod of length \(2 l\) was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable \(X\).

1. Write down the name of the probability density function of \(X\), and specify it fully.
2. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
3. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
4. Find the probability that both of the shorter pieces are of length less than \(\frac { 1 } { 3 } l\).

7. The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} k \left( - x ^ { 2 } + 5 x - 4 \right) , & 1 \leq x \leq 4 \\ 0 , & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 2 } { 9 }\). Find
2. \(\mathrm { E } ( X )\),
3. the mode of \(X\).
4. the cumulative distribution function \(\mathrm { F } ( x )\) for all \(x\).
5. Evaluate \(\mathrm { P } ( X \leq 2.5 )\),
6. Deduce the value of the median and comment on the shape of the distribution.

3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( \mathrm { X } )\),
4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).

5. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k x ( x - 2 ) , & 2 \leq x \leq 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 3 } { 4 }\). Find
2. \(\mathrm { E } ( X )\),
3. the cumulative distribution function \(\mathrm { F } ( x )\).
4. Show that the median value of \(X\) lies between 2.70 and 2.75.

4. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 4,6 ]\).

1. Write down the mean of \(X\).
2. Find \(\mathrm { P } ( X \leqslant 2.4 )\)
3. Find \(\mathrm { P } ( - 3 < X - 5 < 3 )\) The continuous random variable \(Y\) is uniformly distributed over the interval \([ a , 4 a ]\).
4. Use integration to show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }\)
5. Find \(\operatorname { Var } ( Y )\).
6. Given that \(\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)\), find the value of \(a\).

7. The continuous random variable \(X\) has the following probability density function $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(10 a + 25 b = 2\) Given that \(\mathrm { E } ( X ) = \frac { 35 } { 12 }\)
2. find a second equation in \(a\) and \(b\),
3. hence find the value of \(a\) and the value of \(b\).
4. Find, to 3 significant figures, the median of \(X\).
5. Comment on the skewness. Give a reason for your answer.

6. The continuous random variable X has cumulative distribution function \(\mathrm { F } ( x )\) given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 , & x < 1 \\ \frac { 1 } { 27 } \left( - x ^ { 3 } + 6 x ^ { 2 } - 5 \right) , & 1 \leq x \leq 4 \\ 1 , & x > 4 \end{array} \right.$$

1. Find the probability density function \(\mathrm { f } ( x )\).
2. Find the mode of \(X\).
3. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
4. Find the mean \(\mu\) of X .
5. Show that \(\mathrm { F } ( \mu ) > 0.5\).
6. Show that the median of \(X\) lies between the mode and the mean.

7. In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key \(T\) s after the star first appears, a simple model of the game assumes that \(T\) is a continuous uniform random variable defined over the interval \([ 0,1 ]\).

1. Write down \(\mathrm { P } ( \mathrm { T } < 0.2 )\).
2. Write down E(T).
3. Use integration to find \(\operatorname { Var } ( T )\). A group of 20 children each play this game once.
4. Find the probability that no more than 4 children stop the star in less than 0.2 s . The children are allowed to practise.this game so that this continuous uniform model is no longer applicable.
5. Explain how you would expect the mean and variance of T to change. It is found that a more appropriate model of the game when played by experienced players assumes that \(T\) has a probability density function \(\mathrm { g } ( t )\) given by $$g ( t ) = \begin{cases} 4 t , & 0 \leq t \leq 0.5 \\ 4 - 4 t , & 0.5 \leq t \leq 1 , \\ 0 , & \text { otherwise } . \end{cases}$$
6. Using this model show that \(\mathrm { P } ( T < 0.2 ) = 0.08\). A group of 75 experienced players each played this game once.
7. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s .
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2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).

1. Write down the probability density function \(\mathrm { f } ( x )\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( X )\),
4. the cumulative distribution function of \(X\), for all \(x\),
5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).

6. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k \left( 4 x - x ^ { 3 } \right) , & 0 \leqslant x \leqslant 2 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive integer.

1. Show that \(k = \frac { 1 } { 4 }\). Find
2. \(\mathrm { E } ( X )\),
3. the mode of \(X\),
4. the median of \(X\).
5. Comment on the skewness of the distribution.
6. Sketch f(x).

6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).

1. Find \(\mathrm { P } ( A > 3 )\).
2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
3. Find the probability density function of \(Y\).
4. Sketch the probability density function of \(Y\).
5. Write down the mode of \(Y\).
6. Find \(\mathrm { E } ( Y )\).
7. Find \(\mathrm { P } ( Y > 3 )\).

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.
   

4. The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{array} { c c } k \left( 3 + 2 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\)
2. Find the mode of \(X\).
3. Use algebraic integration to find \(\mathrm { E } ( X )\). By comparing your answers to parts (b) and (c),
4. describe the skewness of \(X\), giving a reason for your answer.

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

$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { x ^ { 3 } } { 10 } + \frac { 3 x ^ { 2 } } { 10 } + a x + b & 1 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\).
2. Show that \(\mathrm { f } ( x ) = \frac { 3 } { 10 } \left( x ^ { 2 } + 2 x - 2 \right) , \quad 1 \leqslant x \leqslant 2\)
3. Use integration to find \(\mathrm { E } ( X )\).
4. Show that the lower quartile of \(X\) lies between 1.425 and 1.435

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

1. Sketch f (x).
2. Write down the mode of \(X\). Given that \(\mathrm { E } ( X ) = \frac { 29 } { 16 }\)
3. describe, giving a reason, the skewness of the distribution.
4. Use integration to find the value of \(k\).
5. Write down the lower quartile of \(X\). Given also that \(\mathrm { P } ( 2 < X < 3 ) = \frac { 11 } { 36 }\)
6. find the exact value of \(\mathrm { P } ( X > 3 )\).

7. A piece of string \(A B\) has length 9 cm . The string is cut at random at a point \(P\) and the random variable \(X\) represents the length of the piece of string \(A P\).

1. Write down the distribution of \(X\).
2. Find the probability that the length of the piece of string \(A P\) is more than 6 cm . The two pieces of string \(A P\) and \(P B\) are used to form two sides of a rectangle. The random variable \(R\) represents the area of the rectangle.
3. Show that \(R = a X ^ { 2 } + b X\) and state the values of the constants \(a\) and \(b\).
4. Find \(\mathrm { E } ( R )\).
5. Find the probability that \(R\) is more than twice the area of a square whose side has the length of the piece of string \(A P\).