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The probability generating function of $X$ is $G_x(s)=\frac{1}{2}(s^9(1+s^2))$. Find $EX$ and probability distribution function.

$EX=G_x^{'}(s)=\frac{1}{2}(9s^8+11s^{10})$

How about pdf? Do I need to expand $G_x(s)$ function with Taylor series?

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HINT.

Note that if pdf is $f(k) = \mathbb{P}[X=k]$ for $k \in \mathbb{N}$, then

$$ G(s) = \sum_{k=1}^\infty f(k) s^k = \frac{s^9 \left(1+s^2\right)}{2} = \frac{s^9}{2} + \frac{s^{11}}{2}. $$

That means that $f(k) = 0$ for lots of $k$ -- which ones? And what values does $f(k)$ take where it is non-zero?

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    $\begingroup$ why is $k=1$ in the sum? $\endgroup$ – jay-sun May 21 '13 at 21:00
  • $\begingroup$ @jay-sun Not sure I understand. $X$ maps into $\mathbb{N} = \{1, 2, \ldots \}$, so all elements in $\mathbb{N}$ must be included? $\endgroup$ – gt6989b May 21 '13 at 21:20
  • $\begingroup$ It is obvious from the example that $f(0)=0$, but the general definition of $G(s)$ doesn't need to know that. $\endgroup$ – jay-sun May 21 '13 at 21:25
  • $\begingroup$ @jay-sun The general definition would depend on what you define as the set that $X$ maps to. If you define $Im(f) = \mathbb{N} \cup \{0\}$, then you should include it in the sum. I wrote $f(k) = \mathbb{P}[X=k]$ for $k \in \mathbb{N}$... $\endgroup$ – gt6989b May 21 '13 at 21:31

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