Homework 6.3: Normal approximations (30 pts)

---

a) Imagine I have a univariate continuous distribution with PDF \(f(y)\) that has a maximum at \(y^*\). Assume that the first and second derivatives of \(f(y)\) are defined and continuous near \(y^*\). Show by expanding the log PDF of this distribution in a Taylor series about \(y^*\) that the distribution is locally Normal near the maximum.

In performing the Taylor series, how is the scale parameter \(\sigma\) of the Normal approximation of the distribution related to the log PDF of the distribution is it approximating?

b) Another way you can approximate a distribution as Normal is to use its mean and variance as the parameters as the approximate Normal. We will call this technique “equating moments.” Can you do this if the distribution you are approximating has heavy tails, say like a Cauchy distribution? Why or why not?

c) Make plots of the PDF and CDF of the following distributions with their Normal approximations as derived from the Taylor series and by equating moments. Do you have any comments about the approximations?

• Beta with α = β = 10

• Gamma with α = 5 and β = 2

---