2 The random variable \(X ( X = 1,2,3,4,5,6 )\) denotes the score when a fair six-sided die is rolled.
- Write down the mean of \(X\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\).
- Show that \(\mathrm { G } ( t )\), the probability generating function (pgf) of \(X\), is given by
$$\mathrm { G } ( t ) = \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) }$$
The random variable \(N ( N = 0,1,2 , \ldots )\) denotes the number of heads obtained when an unbiased coin is tossed repeatedly until a tail is first obtained.
- Show that \(\mathrm { P } ( N = r ) = \left( \frac { 1 } { 2 } \right) ^ { r + 1 }\) for \(r = 0,1,2 , \ldots\).
- Hence show that \(\mathrm { H } ( t )\), the pgf of \(N\), is given by \(\mathrm { H } ( t ) = ( 2 - t ) ^ { - 1 }\).
- Use \(\mathrm { H } ( t )\) to find the mean and variance of \(N\).
A game consists of tossing an unbiased coin repeatedly until a tail is first obtained and, each time a head is obtained in this sequence of tosses, rolling a fair six-sided die. The die is not rolled on the first occasion that a tail is obtained and the game ends at that point. The random variable \(Q ( Q = 0,1,2 , \ldots )\) denotes the total score on all the rolls of the die. Thus, in the notation above, \(Q = X _ { 1 } + X _ { 2 } + \ldots + X _ { N }\) where the \(X _ { i }\) are independent random variables each distributed as \(X\), with \(Q = 0\) if \(N = 0\). The pgf of \(Q\) is denoted by \(\mathrm { K } ( t )\). The familiar result that the pgf of a sum of independent random variables is the product of their pgfs does not apply to \(\mathrm { K } ( t )\) because \(N\) is a random variable and not a fixed number; you should instead use without proof the result that \(\mathrm { K } ( t ) = \mathrm { H } ( \mathrm { G } ( t ) )\).
- Show that \(\mathrm { K } ( t ) = 6 \left( 12 - t - t ^ { 2 } - \ldots - t ^ { 6 } \right) ^ { - 1 }\).
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[Hint. \(\left. \left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + \ldots + t ^ { 5 } \right) .\right]\) - Use \(\mathrm { K } ( t )\) to find the mean and variance of \(Q\).
- Using your results from parts (i), (v) and (vii), verify the result that (in the usual notation for means and variances)
$$\sigma _ { Q } { } ^ { 2 } = \sigma _ { N } { } ^ { 2 } \mu _ { X } { } ^ { 2 } + \mu _ { N } \sigma _ { X } { } ^ { 2 } .$$