# Showing that a Gamma distribution converges to a Normal distribution

Consider $$G = \operatorname{Gamma}(p)$$. As $$p$$ goes to $$\infty$$, the Gamma becomes more and more bell-shaped. How do I show that $$\frac{G - p}{\sqrt{p}} \to Z \sim N(0,1)$$ as $$p \to \infty$$?

I started with the CDF of the Gamma and began taking the limit, but it got very messy.

• Have you considered using the MGF? (or the CF) . It's often a convenient strategy. Perhaps consider a Taylor-type expansion. – Glen_b Oct 25 '18 at 5:23
• I have not. My instructor suggested this as a fun practice problem using only the CDF and PDF. – purpleostrich Oct 25 '18 at 5:44
• @StubbornAtom it doesn't help that Z is used to represent two distinct things in the question. It would be necessary to fix that first – Glen_b Oct 25 '18 at 6:00
• The brute-force analysis isn't that difficult if you plan it out. Expand the log of the (unnormalized) PDF of $Z$ in a Maclaurin series. It will equal $$f_Z(z) = -\frac{1}{\sqrt p} + \left(\frac{1}{2p} - \frac{1}{2} \right)z^2 + O(p^{-1/2})O(z^3).$$ Thus its exponential is $e^{-z^2/2}$ times an expression that is very close to $1.$ Justify taking the limit under the integral sign and you're done. – whuber Oct 25 '18 at 14:24
• @Ferdi You're always free to use any resources you like to answer a question. It's a bit of a curiosity that the statement of the present question answers the one on the Math site and the statement of the one on the Math site answers the present question! – whuber Oct 25 '18 at 17:00