Homework 9

Problem 1

Let \(X\sim Gamma(2, 1)\). Verify that the best moment bound is smaller than the best Chernoff bound for \(P(X>10)\).

Problem 2

Let \(X_i\sim U[-1,1]\), \(i=1,...,100\) and independent, let \(\bar{X}=\frac1{100} \sum_{i=1}^{100} X_i\). Find the best Chernoff bound for \(P(|\bar{X}|>0.20)\).

Problem 3

Say \(X_n\sim N(0, \frac1n)\). Show that \(X_n\rightarrow 0\) in quadratic mean, in distribution and in probability using definition 3.2.6 only.