5.03c Calculate mean/variance: by integration

7 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 \\ 1 - \frac { 1 } { x ^ { 4 } } & x \geqslant 1 \end{cases}$$

1. Find the (cumulative) distribution function, \(\mathrm { G } ( y )\), of the random variable \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
2. Hence show that the probability density function of \(Y\) is given by $$g ( y ) = \begin{cases} 2 y & 0 < y \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
3. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).

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 (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 , \\ \frac { 4 } { 3 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 2 , \\ 1 & x > 2 . \end{cases}$$

1. Find the median value of \(X\).
2. Find the (cumulative) distribution function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), and hence find the probability density function of \(Y\).
3. Evaluate \(\mathrm { E } \left( 2 - \frac { 2 } { X ^ { 2 } } \right)\).

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 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.

5 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { ( \alpha - 1 ) ! } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ where \(\alpha\) is a positive integer.

1. Explain how you can deduce that \(\int _ { 0 } ^ { \infty } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } \mathrm {~d} x = ( \alpha - 1 )\) !.
2. Write down an integral for the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) and show, by using the substitution \(x = \frac { u } { 1 - t }\), that \(\mathrm { M } _ { X } ( t ) = ( 1 - t ) ^ { - \alpha }\).
3. Use the moment generating function to find, in terms of \(\alpha\),
   1. \(\mathrm { E } ( X )\),
   2. \(\operatorname { Var } ( X )\).

2 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 4 x e ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that the moment generating function ( mgf ) of \(X\) is $$\frac { 4 } { ( 2 - t ) ^ { 2 } } , \text { where } | t | < 2$$
2. Explain why the \(\operatorname { mgf }\) of \(- X\) is \(\frac { 4 } { ( 2 + t ) ^ { 2 } }\).
3. Two random observations of \(X\) are denoted by \(X _ { 1 }\) and \(X _ { 2 }\). What is the \(\operatorname { mgf }\) of \(X _ { 1 } - X _ { 2 }\) ?

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

1. Show that the moment generating function of \(X\) is \(( 1 - 2 t ) ^ { - 2 }\) for \(t < \frac { 1 } { 2 }\), and state why the condition \(t < \frac { 1 } { 2 }\) is necessary.
2. Use the moment generating function to find \(\operatorname { Var } ( X )\).

4 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } x & 0 \leqslant x \leqslant 1 \\ 2 - x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the moment generating function of \(X\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\). \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent observations of a random variable \(Y\). The moment generating function of \(Y _ { 1 } + Y _ { 2 }\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\).
2. Write down the moment generating function of \(Y\).
3. Use the expansion of \(\mathrm { e } ^ { t }\) to find \(\operatorname { Var } ( Y )\).
4. Deduce the value of \(\operatorname { Var } ( X )\).

2 [In this question, you may use the result \(\int _ { 0 } ^ { \infty } u ^ { m } \mathrm { e } ^ { - u } \mathrm {~d} u = m\) ! for any non-negative integer \(m\).]
The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { \lambda ^ { k + 1 } x ^ { k } \mathrm { e } ^ { - \lambda x } } { k ! } , & x > 0 \\ 0 , & \text { elsewhere } \end{cases}$$ where \(\lambda > 0\) and \(k\) is a non-negative integer.

1. Show that the moment generating function of \(X\) is \(\left( \frac { \lambda } { \lambda - \theta } \right) ^ { k + 1 }\).
2. The random variable \(Y\) is the sum of \(n\) independent random variables each distributed as \(X\). Find the moment generating function of \(Y\) and hence obtain the mean and variance of \(Y\). [8]
3. State the probability density function of \(Y\).
4. For the case \(\lambda = 1 , k = 2\) and \(n = 5\), it may be shown that the definite integral of the probability density function of \(Y\) between limits 10 and \(\infty\) is 0.9165 . Calculate the corresponding probability that would be given by a Normal approximation and comment briefly.

1 The random variable \(X\) has the continuous uniform distribution with probability density function $$\mathrm { f } ( x ) = \frac { 1 } { \theta } , \quad 0 \leqslant x \leqslant \theta$$ where \(\theta ( \theta > 0 )\) is an unknown parameter.
A random sample of \(n\) observations from \(X\) is denoted by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\), with sample mean \(\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).

1. Show that \(2 \bar { X }\) is an unbiased estimator of \(\theta\).
2. Evaluate \(2 \bar { X }\) for a case where, with \(n = 5\), the observed values of the random sample are \(0.4,0.2\), 1.0, 0.1, 0.6. Hence comment on a disadvantage of \(2 \bar { X }\) as an estimator of \(\theta\). For a general random sample of size \(n\), let \(Y\) represent the sample maximum, \(Y = \max \left( X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right)\). You are given that the probability density function of \(Y\) is $$g ( y ) = \frac { n y ^ { n - 1 } } { \theta ^ { n } } , \quad 0 \leqslant y \leqslant \theta$$
3. An estimator \(k Y\) is to be used to estimate \(\theta\), where \(k\) is a constant to be chosen. Show that the mean square error of \(k Y\) is $$k ^ { 2 } \mathrm { E } \left( Y ^ { 2 } \right) - 2 k \theta \mathrm { E } ( Y ) + \theta ^ { 2 }$$ and hence find the value of \(k\) for which the mean square error is minimised.
4. Comment on whether \(k Y\) with the value of \(k\) found in part (iii) suffers from the disadvantage identified in part (ii).

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 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

4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.

1. State what the letter \(x\) represents.
2. Find \(k\) in terms of \(a\).
3. Find \(\operatorname { Var } ( X )\) in terms of \(a\).

3 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { 3 } { 2 a ^ { 3 } } x ^ { 2 } & - a \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(a\) is a constant.

1. It is given that \(\mathrm { P } ( - 3 \leqslant X \leqslant 3 ) = 0.125\). Find the value of \(a\) in this case.
2. It is given instead that \(\operatorname { Var } ( X ) = 1.35\). Find the value of \(a\) in this case.
3. Explain the relationship between \(x\) and \(X\) in this question.

6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k \sin x & 0 \leqslant x \leqslant \frac { 1 } { 2 } \pi , \\ k \left( 2 - \frac { 2 x } { \pi } \right) & \frac { 1 } { 2 } \pi \leqslant x \leqslant \pi , \\ 0 & \text { otherwise, } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 4 + \pi }\).
2. Find \(\mathrm { E } ( X )\), correct to 3 significant figures, showing all necessary working.

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 continuous random variable \(Y\) has a uniform (rectangular) distribution on \([ a , b ]\), where \(a\) and \(b\) are constants.

1. Show that the moment generating function \(\mathrm { M } _ { Y } ( \mathrm { t } )\) of \(Y\) is \(\frac { \left( \mathrm { e } ^ { b t } - \mathrm { e } ^ { a t } \right) } { t ( b - a ) }\).
2. Use the series expansion of \(\mathrm { e } ^ { x }\) to show that the mean and variance of \(Y\) are \(\frac { 1 } { 2 } ( a + b )\) and \(\frac { 1 } { 12 } ( b - a ) ^ { 2 }\), respectively.

5 The discrete random variable \(X\) is such that \(\mathrm { P } ( X = x ) = \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { x } , x = 0,1,2 , \ldots\).

1. Show that the moment generating function of \(X , \mathrm { M } _ { X } ( t )\), can be written as \(\mathrm { M } _ { X } ( t ) = \frac { 3 } { 4 - \mathrm { e } ^ { t } }\).
2. Find the range of values of \(t\) for which the formula for \(\mathrm { M } _ { X } ( t )\) in part (i) is valid.
3. Use \(\mathrm { M } _ { X } ( t )\) to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

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.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { 9 } x ( 3 - x ) & 0 \leqslant x \leqslant 3 , \\ 0 & \text { otherwise } . \end{cases}$$

1. Find the variance of \(X\).
2. Show that the probability that a single observation of \(X\) lies between 0.0 and 0.5 is \(\frac { 2 } { 27 }\).
3. 108 observations of \(X\) are obtained. Using a suitable approximation, find the probability that at least 10 of the observations lie between 0.0 and 0.5 .
4. The mean of 108 observations of \(X\) is denoted by \(\bar { X }\). Write down the approximate distribution of \(\bar { X }\), giving the value(s) of any parameter(s).

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.