Moment generating functions

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

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 \(X _ { 1 } - X _ { 2 }\) ?

2 The random variable \(Z\) has the standard Normal distribution. The random variable \(Y\) is defined by \(Y = Z ^ { 2 }\).
You are given that \(Y\) has the following probability density function. $$\mathrm { f } ( y ) = \frac { 1 } { \sqrt { 2 \pi y } } \mathrm { e } ^ { - \frac { 1 } { 2 } y } , \quad y > 0$$

1. Show that the moment generating function (mgf) of \(Y\) is given by $$\mathrm { M } _ { Y } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } }$$
2. Use the mgf to obtain \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). The random variable \(U\) is defined by $$U = Z _ { 1 } ^ { 2 } + Z _ { 2 } ^ { 2 } + \ldots + Z _ { n } ^ { 2 } ,$$ where \(Z _ { 1 } , Z _ { 2 } , \ldots , Z _ { n }\) are independent standard Normal random variables.
3. State an appropriate general theorem for mgfs and hence write down the mgf of \(U\). State the values of \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\). The random variable \(W\) is defined by $$W = \frac { U - n } { \sqrt { 2 n } }$$
4. Show that the logarithm of the \(\operatorname { mgf }\) of \(W\) is $$- \sqrt { \frac { n } { 2 } } \theta - \frac { n } { 2 } \ln \left( 1 - \sqrt { \frac { 2 } { n } } \theta \right) .$$ Use the series expansion of \(\ln ( 1 - t )\) to show that, as \(n \rightarrow \infty\), this expression tends to \(\frac { 1 } { 2 } \theta ^ { 2 }\).
   State what this implies about the distribution of \(W\) for large \(n\).

1. The random variable \(X\) is such that \(\text{E}(X) = a\theta + b\), where \(a\) and \(b\) are constants and \(\theta\) is a parameter. Show that \(\frac{X - b}{a}\) is an unbiased estimator of \(\theta\). [2]
2. The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{8}(\theta + 4 - x) & \theta \leq x \leq \theta + 4, \\ 0 & \text{otherwise}. \end{cases}$$ Find \(\text{E}(X)\) and hence find an unbiased estimator of \(\theta\). [7]

5 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ and \(k\) is a positive constant.

1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Show that the moment generating function of \(- X\) is \(k ( k + t ) ^ { - 1 }\).
4. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Use the moment generating function of \(X _ { 1 } - X _ { 2 }\) to find the value of \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).

Faults occur in a roll of material at a rate of \(\lambda\) per m\(^2\). To estimate \(\lambda\), three pieces of material of sizes 3 m\(^2\), 7 m\(^2\) and 10 m\(^2\) are selected and the number of faults \(X_1\), \(X_2\) and \(X_3\) respectively are recorded. The estimator \(\hat{\lambda}\), where $$\hat{\lambda} = k(X_1 + X_2 + X_3)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X_1\), \(X_2\) and \(X_3\) and find the value of \(k\). [4]
2. Find Var(\(\hat{\lambda}\)). [3]

A random sample of \(n\) pieces of this material, each of size 4 m\(^2\), was taken. The number of faults on each piece, \(Y\), was recorded.

1. Show that \(\frac{1}{4}\bar{Y}\) is an unbiased estimator of \(\lambda\). [2]
2. Find Var(\(\frac{1}{4}\bar{Y}\)). [3]
3. Find the minimum value of \(n\) for which \(\frac{1}{4}\bar{Y}\) becomes a better estimator of \(\lambda\) than \(\hat{\lambda}\). [2]

3 The moment generating function of a random variable \(X\) is \(( 1 - 2 t ) ^ { - 3 }\).

1. Find the mean and variance of \(X\).
2. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Find \(\mathrm { E } \left[ \left( X _ { 1 } + X _ { 2 } \right) ^ { 3 } \right]\).

1 The random variable \(X\) has the Normal distribution with mean 0 and variance \(\theta\), so that its probability density function is $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 2 \pi \theta } } \mathrm { e } ^ { - x ^ { 2 } / 2 \theta } , \quad - \infty < x < \infty$$ where \(\theta ( \theta > 0 )\) is unknown. A random sample of \(n\) observations from \(X\) is denoted by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\).

1. Find \(\hat { \theta }\), the maximum likelihood estimator of \(\theta\).
2. Show that \(\hat { \theta }\) is an unbiased estimator of \(\theta\).
3. In large samples, the variance of \(\hat { \theta }\) may be estimated by \(\frac { 2 \hat { \theta } ^ { 2 } } { n }\). Use this and the results of parts (i) and (ii) to find an approximate \(95 \%\) confidence interval for \(\theta\) in the case when \(n = 100\) and \(\Sigma X _ { i } ^ { 2 } = 1000\).

A random sample of three independent variables \(X_1\), \(X_2\) and \(X_3\) is taken from a distribution with mean \(\mu\) and variance \(\sigma^2\).

1. Show that \(\frac{2}{5}X_1 - \frac{1}{5}X_2 + \frac{4}{5}X_3\) is an unbiased estimator for \(\mu\). [3]

An unbiased estimator for \(\mu\) is given by \(\hat{\mu} = aX_1 + bX_2\) where \(a\) and \(b\) are constants.

1. Show that Var(\(\hat{\mu}\)) = \((2a^2 - 2a + 1)\sigma^2\). [6]
2. Hence determine the value of \(a\) and the value of \(b\) for which \(\hat{\mu}\) has minimum variance. [5]

6 The random variable \(X\) has a uniform distribution on the interval \([ - 1,1 ]\), so that its probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Show from the definition of the moment generating function that the moment generating function of \(X\) is \(\frac { \sinh t } { t }\).
2. By using the series expansion of \(\sinh t\), find the variance of \(X\) and the value of \(\mathrm { E } \left( X ^ { 4 } \right)\).

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

1. Show that the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) is \(\frac { 4 } { ( 2 - t ) ^ { 2 } }\). You may assume that \(x \mathrm { e } ^ { - k x } \rightarrow 0\) as \(x \rightarrow + \infty\).
2. What condition on \(t\) is needed in finding \(\mathrm { M } _ { X } ( t )\) ?
3. \(Y\) is the sum of three independent observations of \(X\). Find the moment generating function of \(Y\), and use your answer to find \(\operatorname { Var } ( Y )\).

7. \includegraphics[max width=\textwidth, alt={}, center]{65369843-222f-48b2-b8cd-a1c304eac3d9-6_707_718_347_660} The diagram above shows a cyclic quadrilateral \(A B C D\), where \(\widehat { B A D } = \alpha , \widehat { B C D } = \beta\) and \(\alpha + \beta = 180 ^ { \circ }\). These angles are measured.
The random variables \(X\) and \(Y\) denote the measured values, in degrees, of \(\widehat { B A D }\) and \(\widehat { B C D }\) respectively. You are given that \(X\) and \(Y\) are independently normally distributed with standard deviation \(\sigma\) and means \(\alpha\) and \(\beta\) respectively.

1. Calculate, correct to two decimal places, the probability that \(X + Y\) will differ from \(180 ^ { \circ }\) by less than \(\sigma\).
2. Show that \(T _ { 1 } = 45 ^ { \circ } + \frac { 1 } { 4 } ( 3 X - Y )\) is an unbiased estimator for \(\alpha\) and verify that it is a better estimator than \(X\) for \(\alpha\).
3. Now consider \(T _ { 2 } = \lambda X + ( 1 - \lambda ) \left( 180 ^ { \circ } - Y \right)\).
   1. Show that \(T _ { 2 }\) is an unbiased estimator for \(\alpha\) for all values of \(\lambda\).
   2. Find \(\operatorname { Var } \left( T _ { 2 } \right)\) in terms of \(\lambda\) and \(\sigma\).
   3. Hence determine the value of \(\lambda\) which gives the best unbiased estimator for \(\alpha\).

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 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant and \(\theta\) is a parameter taking positive values.

1. Find an expression for \(k\) in terms of \(\theta\).
2. Show that \(\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta\). You are given that \(\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }\). A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) of \(n\) observations of \(X\) is obtained. The estimator \(T _ { 1 }\) is defined as \(T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).
3. Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\), and find the variance of \(T _ { 1 }\).
4. A second unbiased estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)\). For the case \(n = 3\), which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient? \section*{OCR}

2 The random variable \(X\) takes values \(- 2,0\) and 2 , each with probability \(\frac { 1 } { 3 }\).

1. Write down the values of
   (A) \(\mu\), the mean of \(X\),
   (B) \(\mathrm { E } \left( X ^ { 2 } \right)\),
   (C) \(\sigma ^ { 2 }\), the variance of \(X\).
2. Obtain the moment generating function (mgf) of \(X\). A random sample of \(n\) independent observations on \(X\) has sample mean \(\bar { X }\), and the standardised mean is denoted by \(Z\) where $$Z = \frac { \bar { X } - \mu } { \frac { \sigma } { \sqrt { n } } }$$
3. Stating carefully the required general results for mgfs of sums and of linear transformations, show that the mgf of \(Z\) is $$M _ { Z } ( \theta ) = \left\{ \frac { 1 } { 3 } \left( 1 + e ^ { \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } + e ^ { - \frac { \theta \sqrt { 3 } } { \sqrt { 2 n } } } \right) \right\} ^ { n } .$$
4. By expanding the exponential functions in \(\mathrm { M } _ { Z } ( \theta )\), show that, for large \(n\), $$\mathrm { M } _ { Z } ( \theta ) \approx \left( 1 + \frac { \theta ^ { 2 } } { 2 n } \right) ^ { n }$$
5. Use the result \(\mathrm { e } ^ { y } = \lim _ { n \rightarrow \infty } \left( 1 + \frac { y } { n } \right) ^ { n }\) to find the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(n \rightarrow \infty\), and deduce the approximate distribution of \(Z\) for large \(n\).