# Probability mass function (for y)

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.

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