MGF of transformed variable

A question is this type if and only if it asks to find or explain the MGF of a transformed random variable (e.g., -X, aX+b, or X²).

1 questions

OCR MEI S4 2009 June Q2
2
  1. The random variable \(Z\) has the standard Normal distribution with probability density function $$\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - z ^ { 2 } / 2 } , \quad - \infty < z < \infty$$ Obtain the moment generating function of \(Z\).
  2. Let \(\mathrm { M } _ { Y } ( t )\) denote the moment generating function of the random variable \(Y\). Show that the moment generating function of the random variable \(a Y + b\), where \(a\) and \(b\) are constants, is \(\mathrm { e } ^ { b t } \mathrm { M } _ { Y } ( a t )\).
  3. Use the results in parts (i) and (ii) to obtain the moment generating function \(\mathrm { M } _ { X } ( t )\) of the random variable \(X\) having the Normal distribution with parameters \(\mu\) and \(\sigma ^ { 2 }\).
  4. If \(W = \mathrm { e } ^ { X }\) where \(X\) is as in part (iii), \(W\) is said to have a lognormal distribution. Show that, for any positive integer \(k\), the expected value of \(W ^ { k }\) is \(\mathrm { M } _ { X } ( k )\). Use this result to find the expected value and variance of the lognormal distribution.