Probability with Measure

6.4 Exercises on Chapter 6 \((\Delta )\)

6.1 Let \(X_n\) have a Poisson distribution with mean \(n\lambda >0\). Show that \(\E [e^{tX_n}]=e^{n\lambda (e^t-1)}\). Hence construct a Chernoff bound for \(\P [X_n>n\lambda ^2]\), where \(n\in \N \).
6.2 Let \((E_n)_{n\in \N }\) be a sequence of events and let \(X_n=\sum _{i=1}^n \1_{E_i}\). Suppose \(\sum _{i=1}^\infty \P [E_i]=\infty \) and that \(\P [E_i \cap E_j]\leq \P [E_i]\P [E_j]\) for all \(i\neq j\). Show that \(\P [X_n\geq 1]\to 1\) as \(n\to \infty \).
6.3 In each case, determine whether the two quantities given satisfy an equality of the form \(a\leq b\), \(b\leq a\), or if no such inequality holds in general.
- (a) \(\E [X^{4}]\) and \(\E [X]^{4}\), where \(X\) is a random variable.
- (b) \(\E [X^{1/4}]\) and \(\E [X]^{1/4}\), where \(X\) is a non-negative random variable.
- (c) \(\E [e^X]\) and \(e^{\E [X]}\), where \(X\) is a bounded random variable.
- (d) \(\E [\cos (X)]\) and \(\cos (\E [X])\), where \(X\) is a random variable.
6.4 Let \(\{x_1,\ldots ,x_n\}\sw (0,\infty )\). The mean average of these values is \(\frac {x_1+\ldots x_n}{n}\), which is more precisely known as the arithmetic mean. In some situations it is advantageous to instead use the geometric mean, \(\sqrt [n]{x_1x_2\ldots x_n}\).

By applying Jensen’s inequality to a random variable with the discrete uniform distribution on \(\{x_1,\ldots ,x_m\}\) and the function \(g(x)=-\log x\), deduce that

\[\sqrt [n]{x_1x_2\ldots x_n} \leq \frac {x_1+\ldots x_n}{n}.\]

This equation is known as the AM-GM inequality.
6.5 Let \(1\leq p\leq q\) and let \(X\) be a random variable. Use Jensen’s inequality to show that if \(\E [|X|^q]<\infty \) then \(\E [|X|^p]<\infty \).

Challenge questions

6.6 Let \(X\) be a non-negative random variable with \(\E [X^2]\in (0,\infty )\). Show that \(\E [X]>0\) and that

\[\P [X=0]\leq \min \l \{\frac {\E [X^2]}{\E [X]^2}-1,\;1-\frac {\E [X]^2}{\E [X^2]}\r \}.\]