91Ó°ÊÓ

If \(X\) is an integer-valued random variable with characteristic function \(\phi\), show that $$ P(X=k)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{-i t k} \phi(t) d t $$ What is the corresponding result for a random variable whose distribution is arithmetic with span \(\lambda\) (that is, there is probability one that \(X\) is a multiple of \(\lambda\), and \(\lambda\) is the largest positive number with this? property)?

Find the characteristic function of \(X^{2}\) when \(X\) has the \(N\left(\mu, \sigma^{2}\right)\) distribution.

For the simple random walk, show that the probability \(p_{0}(2 n)\) that the particle retums to the origin at the \((2 n)\) th step satisfies \(p_{0}(2 n) \sim(4 p q)^{n} / \sqrt{\pi n}\), and use this to prove that the walk is persistent if and only if \(p=\frac{1}{2}\), You will need Stirling's formula: \(n ! \sim n^{n+\frac{1}{2}} e^{-n} \sqrt{2 \pi}\)

Find the characteristic functions of the following density functions: (a) \(f(x)=\frac{1}{2} e^{-|x|}\) for \(x \in R\) (b) \(f(x)=\frac{1}{2}|x| e^{-|x|}\) for \(x \in \mathbb{R}\)

The distribution of a random variable \(X\) is called infinitely divisible if, for all positive integers \(n\), there exists a sequence \(Y_{1}^{(n)}, Y_{2}^{(n)} \ldots ., Y_{n}^{(n)}\) of independent identically distributed random variables such that \(X\) and \(Y_{1}^{(n)}+Y_{2}^{(n)}+\cdots+Y_{n}^{(n)}\) have the same distribution. (a) Show that the normal, Poisson, and gamma distributions are infinitely divisible. (b) Show that the characteristic function \(\phi\) of an infinitely divisible distribution has no real zeros, in that \(\phi(t) \neq 0\) for all real \(t\).

A random variable \(X\) is called symmetric if \(X\) and \(-X\) are identically distributed. Show that \(X\) is symmetric if and only if the imaginary part of its characteristic function is identically zero.