5.03b Solve problems: using pdf

5 Two random variables \(S\) and \(T\) have probability density functions given by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { 3 } { a ^ { 3 } } ( x - a ) ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} c & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where \(a\) and \(c\) are constants.

1. On a single diagram sketch both probability density functions.
2. Calculate the mean of \(S\), in terms of \(a\).
3. Use your diagram to explain which of \(S\) or \(T\) has the bigger variance. (Answers obtained by calculation will score no marks.)

4 The weekly sales of petrol, \(X\) thousand litres, at a garage may be modelled by a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} c & 25 \leqslant x \leqslant 45 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant. The weekly profit, in \(\pounds\), is given by \(( 400 \sqrt { X } - 240 )\).

1. Obtain the value of \(c\).
2. Find the expected weekly profit.
3. Find the probability that the weekly profit exceeds \(\pounds 2000\).

1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 2 } { 5 } & - a \leqslant x < 0 \\ \frac { 2 } { 5 } \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$ Find

1. the value of the constant \(a\),
2. \(\mathrm { E } ( X )\).

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \sqrt { x } & 0 < x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ The random variable \(Y\) is given by \(Y = \frac { 1 } { \sqrt { X } }\).

1. Find the (cumulative) distribution function of \(Y\), and hence show that its probability density function is given by $$\mathrm { g } ( y ) = \frac { 3 } { y ^ { 4 } }$$ for a set of values of \(y\) to be stated.
2. Find the value of \(\mathrm { E } \left( Y ^ { 2 } \right)\).

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

1. \(\mathrm { E } ( X )\),
2. \(\mathrm { P } ( X \geqslant \mathrm { E } ( X ) )\).

6 The function \(\mathrm { F } ( t )\) is defined as follows. $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 \\ \sin ^ { 4 } t & 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi \\ 1 & t > \frac { 1 } { 2 } \pi \end{cases}$$

1. Verify that F is a (cumulative) distribution function. The continuous random variable \(T\) has (cumulative) distribution function \(\mathrm { F } ( t )\).
2. Find the lower quartile of \(T\).
3. Find the (cumulative) distribution function of \(Y\), where \(Y = \sin T\), and obtain the probability density function of \(Y\).
4. Find the expected value of \(\frac { 1 } { Y ^ { 3 } + 2 Y ^ { 4 } }\).

4 A timber supplier cuts wooden fence posts from felled trees. The posts are of length \(( k + X ) \mathrm { cm }\) where \(k\) is a constant and \(X\) is a random variable which has probability density function $$f ( x ) = \begin{cases} 1 + x & - 1 \leqslant x < 0 \\ 1 - x & 0 \leqslant x \leqslant 1 \\ 0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Write down the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Write down, in terms of \(k\), the approximate distribution of \(\bar { L }\), the mean length of a random sample of 50 fence posts. Justify your choice of distribution.
4. In a particular sample of 50 posts, the mean length is 90.06 cm . Find a \(95 \%\) confidence interval for the true mean length of the fence posts.
5. Explain whether it is reasonable to suppose that \(k = 90\).

4 At the school summer fair, one of the games involves throwing darts at a circular dartboard of radius \(a\) lying on the ground some distance away. Only darts that land on the board are counted. The distance from the centre of the board to the point where a dart lands is modelled by the random variable \(R\). It is assumed that the probability that a dart lands inside a circle of radius \(r\) is proportional to the area of the circle.

1. By considering \(\mathrm { P } ( R < r )\) show that \(\mathrm { F } ( r )\), the cumulative distribution function of \(R\), is given by $$\mathrm { F } ( r ) = \begin{cases} 0 & r < 0 , \\ \frac { r ^ { 2 } } { a ^ { 2 } } & 0 \leqslant r \leqslant a , \\ 1 & r > a . \end{cases}$$
2. Find \(\mathrm { f } ( r )\), the probability density function of \(R\).
3. Find \(\mathrm { E } ( R )\) and show that \(\operatorname { Var } ( R ) = \frac { a ^ { 2 } } { 18 }\). The radius \(a\) of the dartboard is 22.5 cm .
4. Let \(\bar { R }\) denote the mean distance from the centre of the board of a random sample of 100 darts. Write down an approximation to the distribution of \(\bar { R }\).
5. A random sample of 100 darts is found to give a mean distance of 13.87 cm . Does this cast any doubt on the modelling?

2 A particular species of reed that grows up to 2 metres in length is used for thatching. The lengths in metres of the reeds when harvested are modelled by the random variable \(X\) which has the following probability density function, \(\mathrm { f } ( x )\). $$f ( x ) = \begin{cases} \frac { 3 } { 16 } \left( 4 x - x ^ { 2 } \right) & \text { for } 0 \leqslant x \leqslant 2 \\ 0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { E } ( X ) = \frac { 5 } { 4 }\) and find the standard deviation of the lengths of the harvested reeds.
3. Find the standard error of the mean length for a random sample of 100 reeds. Once the harvested reeds have been collected, any that are shorter than 1 metre are discarded.
4. Find the proportion of reeds that should be discarded according to the model.
5. Reeds are harvested from a large area which is divided into several reed beds. A sample of the harvested reeds is required for quality control. How might the method of cluster sampling be used to obtain it?

4 A random variable \(X\) has probability density function \(\mathrm { f } ( x ) = \frac { 2 x } { \lambda ^ { 2 } }\) for \(0 < x < \lambda\), where \(\lambda\) is a positive constant.

1. Show that, for any value of \(\lambda , \mathrm { f } ( x )\) is a valid probability density function.
2. Find \(\mu\), the mean value of \(X\), in terms of \(\lambda\) and show that \(\mathrm { P } ( X < \mu )\) does not depend on \(\lambda\).
3. Given that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { \lambda ^ { 2 } } { 2 }\), find \(\sigma ^ { 2 }\), the variance of \(X\), in terms of \(\lambda\). The random variable \(X\) is used to model the depth of the space left by the filling machine at the top of a jar of jam. The model gives the following probabilities for \(X\) (whatever the value of \(\lambda\) ).

   \(0 < X \leqslant \mu - \sigma\)	\(\mu - \sigma < X \leqslant \mu\)	\(\mu < X \leqslant \mu + \sigma\)	\(\mu + \sigma < X < \lambda\)
   0.18573	0.25871	0.36983	0.18573

   A sample of 50 random observations of \(X\), classified in the same way, is summarised by the following frequencies.

   4

   11

   20

   15

4. Carry out a suitable test at the \(5 \%\) level of significance to assess the goodness of fit of \(X\) to these data. Explain briefly how your conclusion may be affected by the choice of significance level.

3 The random variable \(X\) has the following probability density function, \(\mathrm { f } ( x )\). $$f ( x ) = \begin{cases} k x ( x - 5 ) ^ { 2 } & 0 \leqslant x < 5 \\ 0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Find, in terms of \(k\), the cumulative distribution function, \(\mathrm { F } ( x )\).
3. Hence show that \(k = \frac { 12 } { 625 }\). The random variable \(X\) is proposed as a model for the amount of time, in minutes, lost due to stoppages during a football match. The times lost in a random sample of 60 matches are summarised in the table. The table also shows some of the corresponding expected frequencies given by the model.

   Time (minutes)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
   Observed frequency	5	15	23	11	6
   Expected frequency			17.76	9.12	1.632

4. Find the remaining expected frequencies.
5. Carry out a goodness of fit test, using a significance level of \(2.5 \%\), to see if the model might be suitable in this context.

4 The probability density function of a random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant a \\ k ( 2 a - x ) & a < x \leqslant 2 a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(k\) are positive constants.

1. Sketch \(\mathrm { f } ( x )\). Hence explain why \(\mathrm { E } ( X ) = a\).
2. Show that \(k = \frac { 1 } { a ^ { 2 } }\).
3. Find \(\operatorname { Var } ( X )\) in terms of \(a\). In order to estimate the value of \(a\), a random sample of size 50 is taken from the distribution. It is found that the sample mean and standard deviation are \(\bar { x } = 1.92\) and \(s = 0.8352\).
4. Construct a symmetrical \(95 \%\) confidence interval for \(a\). Give one reason why the answer is only approximate.
5. A non-statistician states that the probability that \(a\) lies in the interval found in part (iv) is 0.95 . Comment on this statement. \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {® } }\)}

3 The random variable \(X\) has the following probability density function: $$\mathrm { f } ( x ) = \begin{cases} k \left( 1 - x ^ { 2 } \right) & - 1 \leqslant x \leqslant 1 \\ 0 & \text { elsewhere } \end{cases}$$ where \(k\) is a positive constant.

1. Calculate the value of \(k\).
2. Sketch the probability density function.
3. Calculate \(\operatorname { Var } ( X )\).
4. Find a cubic equation satisfied by the upper quartile \(q\), and hence verify that \(q = 0.35\) to 2 decimal places.
5. A random sample of 40 values of \(X\) is taken. Using a suitable approximating distribution, calculate the probability that the mean of these values is greater than 0.125 . Justify your choice of distribution.

3 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \mathrm { e } ^ { 2 x } & x < 0 \\ \mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$

1. Show that the moment generating function of \(X\) is \(\frac { 4 } { 4 - t ^ { 2 } }\), where \(| t | < 2\), and explain why the condition \(| t | < 2\) is necessary.
2. Find \(\operatorname { Var } ( X )\).

6 Nuts and raisins occur in randomly chosen squares of a particular brand of chocolate. The numbers of nuts and raisins are denoted by \(N\) and \(R\) respectively and the joint probability distribution of \(N\) and \(R\) is given by $$f ( n , r ) = \begin{cases} c ( n + 2 r ) & n = 0,1,2 \text { and } r = 0,1,2 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.

1. Find the value of \(c\).
2. Find the probability that there is exactly one nut in a randomly chosen square.
3. Find the probability that the total number of nuts and raisins in a randomly chosen square is more than 2 .
4. For squares in which there are 2 raisins, find the mean number of nuts.
5. Determine whether \(N\) and \(R\) are independent.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is an unknown positive constant.

1. Find \(\mathrm { E } \left( X ^ { n } \right)\), where \(n \neq - 2\), and hence write down the value of \(\mathrm { E } ( X )\).
2. Find
   1. \(\operatorname { Var } ( X )\),
   2. \(\operatorname { Var } \left( X ^ { 2 } \right)\).
   3. Find \(\mathrm { E } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)\) and \(\mathrm { E } \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } + X _ { 3 } ^ { 2 } \right)\), where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\). Hence construct unbiased estimators, \(T _ { 1 }\) and \(T _ { 2 }\), of \(\theta\) and \(\operatorname { Var } ( X )\) respectively, which are based on \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
   4. Find \(\operatorname { Var } \left( T _ { 2 } \right)\).

7 The discrete random variable \(X\) can take the values 0,1 and 2 with equal probabilities.
The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\), and the random variables \(Y\) and \(Z\) are defined as follows: \(Y\) is the smaller of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal; \(Z = \left| X _ { 1 } - X _ { 2 } \right|\).

1. Draw up a table giving the joint distribution of \(Y\) and \(Z\).
2. Find \(P ( Y = 0 \mid Z = 0 )\).
3. Find \(\operatorname { Cov } ( Y , Z )\).

7 Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of a continuous random variable with probability density function $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a parameter whose value is to be estimated.

1. Find \(\mathrm { E } ( X )\).
2. Show that \(S _ { 1 } = X _ { 1 } + X _ { 2 }\) is an unbiased estimator of \(\theta\). \(L\) is the larger of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal.
3. Show that the probability density function of \(L\) is \(\frac { 2 l } { \theta ^ { 2 } }\) for \(0 \leqslant l \leqslant \theta\).
4. Find \(\mathrm { E } ( L )\).
5. Find an unbiased estimator \(S _ { 2 }\) of \(\theta\), based on \(L\).
6. Determine which of the two estimators \(S _ { 1 }\) and \(S _ { 2 }\) is the more efficient.

7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ 1 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \end{cases}$$ For a random value of \(X\), find the probability that 2 lies between \(X\) and \(4 X\). Find also the expected value of the width of the interval ( \(X , 4 X\) ).

The continuous random variable \(T\) has probability density function given by $$\mathrm { f } ( t ) = \begin{cases} 0 & t < 2 \\ \frac { 2 } { ( t - 1 ) ^ { 3 } } & t \geqslant 2 \end{cases}$$

1. Find the distribution function of \(T\), and find also \(\mathrm { P } ( T > 5 )\).
2. Consecutive independent observations of \(T\) are made until the first observation that exceeds 5 is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding 5. Find \(\mathrm { P } ( N > \mathrm { E } ( N ) )\).
3. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { T - 1 }\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

5 The continuous random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.01 \mathrm { e } ^ { - 0.01 x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$

1. State the value of \(\mathrm { E } ( X )\).
2. Find the median value of \(X\).
3. Find the probability that \(X\) lies between the median and the mean.

7 The waiting time, \(T\) minutes, before a customer is served in a restaurant has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \\ 0 & t < 0 \end{cases}$$ where \(\lambda\) is a positive constant. The standard deviation of \(T\) is 8 . Find

1. the value of \(\lambda\),
2. the probability that a customer has to wait between 5 and 10 minutes before being served,
3. the median value of \(T\).

6 The random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.6 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ Identify the distribution of \(X\) and state its mean. Find

1. \(\mathrm { P } ( X > 4 )\),
2. the median of \(X\).