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The probability mass function of a random variable $Y$ is given by

$$f(y) = \frac{a^{(y-3)}}{(e^a)(y-3)!} \quad \text{for } y = 3,4,\dots\text{ and } a > 0$$

Derive the cumulant generating function of $Y$ and use it to obtain the kurtosis of $Y$

I think you need to use some sort of substitution to transform this into a pmf (eg $x = y - 3$) but I'm just not sure how exactly to account for the substitution.

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  • $\begingroup$ Gamma? This is a discrete distribution. Why don't you just apply the definitions? $\endgroup$ – StubbornAtom May 9 at 17:48
  • $\begingroup$ Thank you but i'm still confused as to how to proceed $\endgroup$ – KombatWombat May 9 at 17:52
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    $\begingroup$ Are you asking for the definitions or are you stuck somewhere? Please show your attempt. $\endgroup$ – StubbornAtom May 9 at 17:56
  • $\begingroup$ I'm asking am I not supposed to use some substitution to transform this into a known distribution then derive the cgf from there? $\endgroup$ – KombatWombat May 9 at 18:02
  • $\begingroup$ Try the Poisson distribution. $\endgroup$ – Brian Tung May 9 at 18:07

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