5.03b Solve problems: using pdf

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.

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 , \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(k\) is a constant.

1. Sketch the graph of f.
2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-09_2725_35_99_20}
3. Find the cumulative distribution function of \(X\).
4. Find the median value of \(X\).

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Sketch the graph of f.
2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-09_2725_35_99_20}
3. Find the cumulative distribution function of \(X\).
4. Find the median value of \(X\).

7 A continuous random variable \(X _ { 1 }\) has probability density function given by $$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
4. Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
5. The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).

7 The continuous random variable \(X\) has the probability density function shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}

1. Find the value of the constant \(k\).
2. Write down the mean of \(X\), and use integration to find the variance of \(X\).
3. Three observations of \(X\) are made. Find the probability that \(X < 9\) for all three observations.
4. The mean of 32 observations of \(X\) is denoted by \(\bar { X }\). State the approximate distribution of \(\bar { X }\), giving its mean and variance. \section*{[Question 8 is printed overleaf.]}

7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$

1. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
2. Explain in everyday terms the difference between the two random variables.
3. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\).

7 The time, in minutes, for which a customer is prepared to wait on a telephone complaints line is modelled by the random variable \(X\). The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 81 }\).
2. Find \(\mathrm { E } ( X )\).
3. (a) Show that the value \(y\) which satisfies \(\mathrm { P } ( X < y ) = \frac { 3 } { 5 }\) satisfies $$5 y ^ { 4 } - 90 y ^ { 2 } + 243 = 0 .$$ (b) Using the substitution \(w = y ^ { 2 }\), or otherwise, solve the equation in part (a) to find the value of \(y\).

3 For a restaurant with a home-delivery service, the delivery time in minutes can be modelled by a continuous random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { \pi } { 90 } \sin \left( \frac { \pi t } { 60 } \right) & 20 \leqslant t \leqslant 60 \\ 0 & \text { otherwise. } \end{cases}$$

1. Given that \(20 \leqslant a \leqslant 60\), show that \(\mathrm { P } ( T \leqslant a ) = \frac { 1 } { 3 } \left( 1 - 2 \cos \left( \frac { \pi a } { 60 } \right) \right)\). There is a delivery charge of \(\pounds 3\) but this is reduced to \(\pounds 2\) if the delivery time exceeds a minutes.
2. Find the value of \(a\) for which the expected value of the delivery charge for a home-delivery is £2.80.

7 The continuous random variable \(T\) has probability density function given by $$f ( t ) = \begin{cases} 4 t ^ { 3 } & 0 < t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Obtain the cumulative distribution function of \(T\).
2. Find the cumulative distribution function of \(H\), where \(H = \frac { 1 } { T ^ { 4 } }\), and hence show that the probability density function of \(H\) is given by \(\mathrm { g } ( h ) = \frac { 1 } { h ^ { 2 } }\) over an interval to be stated.
3. Find \(\mathrm { E } \left( 1 + 2 H ^ { - 1 } \right)\).

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leqslant x < 2 \\ \frac { 1 } { 12 } x & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the upper quartile of \(X\).
2. Find the value of \(a\) for which \(\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )\).

3 The monthly demand for a product, \(X\) thousand units, is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1 \\ a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find

1. the value of \(a\),
2. the probability that the monthly demand is at most 1500 units,
3. the expected monthly demand.

5 The continuous random variable \(X\) has a triangular distribution with probability density function given by $$f ( x ) = \left\{ \begin{array} { l r } 1 + x & - 1 \leqslant x \leqslant 0 \\ 1 - x & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that, for \(0 \leqslant a \leqslant 1\), $$\mathrm { P } ( | X | \leqslant a ) = 2 a - a ^ { 2 } .$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
2. Express \(\mathrm { P } ( Y \leqslant y )\) in terms of \(y\), for \(0 \leqslant y \leqslant 1\), and hence show that the probability density function of \(Y\) is given by $$g ( y ) = \frac { 1 } { \sqrt { } y } - 1 , \quad \text { for } 0 < y \leqslant 1 .$$
3. Use the probability density function of \(Y\) to find \(\mathrm { E } ( Y )\), and show how the value of \(\mathrm { E } ( Y )\) may also be obtained directly using the probability density function of \(X\).
4. Find \(\mathrm { E } ( \sqrt { } Y )\).

1 The continuous random variable \(X\) has probability density function $$f ( x ) = k ( 1 - x ) \quad \text { for } 0 \leqslant x \leqslant 1$$ where \(k\) is a constant.

1. Show that \(k = 2\). Sketch the graph of the probability density function.
2. Find \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 1 } { 18 }\).
3. Derive the cumulative distribution function of \(X\). Hence find the probability that \(X\) is greater than the mean.
4. Verify that the median of \(X\) is \(1 - \frac { 1 } { \sqrt { 2 } }\).
5. \(\bar { X }\) is the mean of a random sample of 100 observations of \(X\). Write down the approximate distribution of \(\bar { X }\).

1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable \(X\) having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$

1. Find the mean and the mode of \(X\).
2. Find the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\). $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.

   Load \(x\)	\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)	\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)	\(\frac { 3 } { 4 } < x \leqslant 1\)
   Frequency	126	209	131	46

3. Use a suitable statistical procedure to assess the goodness of fit of \(X\) to these data. Discuss your conclusions briefly.

1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$ where \(k\) is a constant.

1. Show that \(k = \frac { 5 } { 8 }\).
2. Find the modal time.
3. Find \(\mathrm { E } ( T )\) and show that \(\operatorname { Var } ( T ) = \frac { 8 } { 63 }\).
4. A large random sample of \(n\) fireworks of this type is tested. Write down in terms of \(n\) the approximate distribution of \(\bar { T }\), the sample mean time.
5. For a random sample of 100 such fireworks the times are summarised as follows. $$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$ Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.

1 The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \frac { x \mathrm { e } ^ { - x / \lambda } } { \lambda ^ { 2 } } \quad ( x > 0 )$$ where \(\lambda\) is a parameter \(( \lambda > 0 ) . X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) are \(n\) independent observations on \(X\), and \(\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\) is their mean.

1. Obtain \(\mathrm { E } ( X )\) and deduce that \(\hat { \lambda } = \frac { 1 } { 2 } \bar { X }\) is an unbiased estimator of \(\lambda\).
2. \(\operatorname { Obtain } \operatorname { Var } ( \hat { \lambda } )\).
3. Explain why the results in parts (i) and (ii) indicate that \(\hat { \lambda }\) is a good estimator of \(\lambda\) in large samples.
4. Suppose that \(n = 3\) and consider the alternative estimator $$\tilde { \lambda } = \frac { 1 } { 8 } X _ { 1 } + \frac { 1 } { 4 } X _ { 2 } + \frac { 1 } { 8 } X _ { 3 } .$$ Show that \(\tilde { \lambda }\) is an unbiased estimator of \(\lambda\). Find the relative efficiency of \(\tilde { \lambda }\) compared with \(\hat { \lambda }\). Which estimator do you prefer in this case?

1 In a certain country, any baby born is equally likely to be a boy or a girl, independently for all births. The birthweight of a baby boy is given by the continuous random variable \(X _ { B }\) with probability density function (pdf) \(\mathrm { f } _ { B } ( x )\) and cumulative distribution function (cdf) \(\mathrm { F } _ { B } ( x )\). The birthweight of a baby girl is given by the continuous random variable \(X _ { G }\) with pdf \(\mathrm { f } _ { G } ( x )\) and cdf \(\mathrm { F } _ { G } ( x )\). The continuous random variable \(X\) denotes the birthweight of a baby selected at random.

1. By considering $$\mathrm { P } ( X \leqslant x ) = \mathrm { P } ( X \leqslant x \mid \text { boy } ) \mathrm { P } ( \text { boy } ) + \mathrm { P } ( X \leqslant x \mid \text { girl } ) \mathrm { P } ( \text { girl } ) ,$$ find the cdf of \(X\) in terms of \(\mathrm { F } _ { B } ( x )\) and \(\mathrm { F } _ { G } ( x )\), and deduce that the pdf of \(X\) is $$\mathrm { f } ( x ) = \frac { 1 } { 2 } \left\{ \mathrm { f } _ { B } ( x ) + \mathrm { f } _ { G } ( x ) \right\} .$$
2. The birthweights of baby boys and girls have means \(\mu _ { B }\) and \(\mu _ { G }\) respectively. Deduce that $$\mathrm { E } ( X ) = \frac { 1 } { 2 } \left( \mu _ { B } + \mu _ { G } \right) .$$
3. The birthweights of baby boys and girls have common variance \(\sigma ^ { 2 }\). Find an expression for \(\mathrm { E } \left( X ^ { 2 } \right)\) in terms of \(\mu _ { B } , \mu _ { G }\) and \(\sigma ^ { 2 }\), and deduce that $$\operatorname { Var } ( X ) = \sigma ^ { 2 } + \frac { 1 } { 4 } \left( \mu _ { B } - \mu _ { G } \right) ^ { 2 } .$$
4. A random sample of size \(2 n\) is taken from all the babies born in a certain period. The mean birthweight of the babies in this sample is \(\bar { X }\). Write down an approximation to the sampling distribution of \(\bar { X }\) if \(n\) is large.
5. Suppose instead that a stratified sample of size \(2 n\) is taken by selecting \(n\) baby boys at random and, independently, \(n\) baby girls at random. The mean birthweight of the \(2 n\) babies in this sample is \(\bar { X } _ { s t }\). Write down the expected value of \(\bar { X } _ { s t }\) and find the variance of \(\bar { X } _ { s t }\).
6. Deduce that both \(\bar { X }\) and \(\bar { X } _ { s t }\) are unbiased estimators of the population mean birthweight. Find which is the more efficient.

1 The random variable \(X\) has the following probability density function, in which \(a\) is a (positive) parameter. $$\mathrm { f } ( x ) = \frac { 2 } { a } x \mathrm { e } ^ { - x ^ { 2 } / a } , \quad x \geqslant 0 .$$

1. Verify that \(\int _ { 0 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x = 1\).
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = a\) and \(\mathrm { E } \left( X ^ { 4 } \right) = 2 a ^ { 2 }\). The parameter \(a\) is to be estimated by maximum likelihood based on an independent random sample from the distribution, \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\).
3. Show that the logarithm of the likelihood function is $$n \ln 2 - n \ln a + \sum _ { i = 1 } ^ { n } \ln X _ { i } - \frac { 1 } { a } \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 }$$ Hence obtain the maximum likelihood estimator, \(\hat { a }\), for \(a\).
   [0pt] [You are not required to verify that any turning point you find is a maximum.]
4. Using the results from part (ii), show that \(\hat { a }\) is unbiased for \(a\) and find the variance of \(\hat { a }\).
5. In a particular random sample from this distribution, \(n = 100\) and \(\sum x _ { i } ^ { 2 } = 147.1\). Obtain an approximate 95\% confidence interval for \(a\). (You may assume that the Central Limit Theorem holds in this case.) Option 2: Generating Functions

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } k x & 0 \leqslant x < 2 \\ \frac { k ( 4 - x ) ^ { 2 } } { 2 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 10 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find the cumulative distribution function of \(X\).
4. Find the upper quartile of \(X\), correct to 3 significant figures. \section*{END OF QUESTION PAPER}

4 The number, \(x\), of a certain type of sea shell was counted at 60 randomly chosen sites, each one metre square, along the coastline in country \(A\). The number, \(y\), of the same type of sea shell was counted at 50 randomly chosen sites, each one metre square, along the coastline in country \(B\). The results are summarised as follows, where \(\bar{x}\) and \(\bar{y}\) denote the sample means of \(x\) and \(y\) respectively. $$\bar{x} = 29.2 \quad \Sigma(x - \bar{x})^{2} = 4341.6 \quad \bar{y} = 24.4 \quad \Sigma(y - \bar{y})^{2} = 3732.0$$ Find a \(95\%\) confidence interval for the difference between the mean number of sea shells, per square metre, on the coastlines in country \(A\) and in country \(B\).

5 The continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & \mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \\ T : & \mathrm { g } ( x ) = \begin{cases} \frac { 5 } { 64 } x ^ { 4 } & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases} \end{array}$$

1. Sketch, on the same axes, the graphs of f and g .
2. Describe in everyday terms the difference between the distributions of the random variables \(S\) and \(T\). (Answers that comment only on the shapes of the graphs will receive no credit.)
3. Calculate the variance of \(T\).

7 Two continuous random variables \(S\) and \(T\) have probability density functions \(\mathrm { f } _ { S }\) and \(\mathrm { f } _ { T }\) given respectively by $$\begin{aligned} & f _ { S } ( x ) = \begin{cases} \frac { a } { x ^ { 2 } } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \\ & f _ { T } ( x ) = \begin{cases} b & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases} \end{aligned}$$ where \(a\) and \(b\) are constants.

1. Sketch on the same axes the graphs of \(y = \mathrm { f } _ { S } ( x )\) and \(y = \mathrm { f } _ { T } ( x )\).
2. Find the value of \(a\).
3. Find \(\mathrm { E } ( S )\).
4. A student gave the following description of the distribution of \(T\) : "The probability that \(T\) occurs is constant". Give an improved description, in everyday terms.

8 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} k x ^ { - a } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants and \(a\) is greater than 1 .

1. Show that \(k = a - 1\).
2. Find the variance of \(X\) in the case \(a = 4\).
3. It is given that \(\mathrm { P } ( X < 2 ) = 0.9\). Find the value of \(a\), correct to 3 significant figures.

4 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Calculate the variance of \(X\).
3. A student writes " \(X\) is more likely to occur when \(x\) takes values further away from 2 ". Explain whether you agree with this statement.

7 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} k x ^ { 2 } & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(k\) are constants.

1. Sketch the graph of \(y = \mathrm { f } ( x )\) and explain in non-technical language what this tells you about \(X\).
2. Given that \(\mathrm { E } ( X ) = 4.5\), find
   1. the value of \(a\),
   2. \(\operatorname { Var } ( X )\).