Derive MGF from PDF

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 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 \(n\).

2 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x } , \quad x > 0 .$$

1. Obtain the moment generating function (mgf) of \(X\).
2. Use the mgf to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). The random variable \(Y\) is defined as follows: $$Y = X _ { 1 } + \ldots + X _ { n } ,$$ where the \(X _ { i }\) are independently and identically distributed as \(X\).
3. Write down expressions for \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). Obtain the \(\operatorname { mgf }\) of \(Y\).
4. Find the \(\operatorname { mgf }\) of \(Z\) where \(Z = \frac { Y - \frac { n } { \lambda } } { \frac { \sqrt { n } } { \lambda } }\).
5. By considering the logarithm of the mgf of \(Z\), show that the distribution of \(Z\) tends to the standard Normal distribution as \(n\) tends to infinity.

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.

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