Prove that the distributions of

(a) Pascal (Exercise 1 (a) of Chapter 5),

(b) Polya (Exercise 1 (b) of Chapter 5),

(c) Cauchy (Example 5, Ssc. 24)

are infinitely divisible.

Exercise 1:

(a) \(\mathrm{P}(\xi=k)=\frac{a^{k}}{(1+a)^{k+1}}, a>0\) is a constant (this is the Pascal distribution).

(b) \(p_{k}=\mathbf{P}\{\xi=k\}=\left(\frac{\alpha \lambda}{1+\alpha \lambda}\right)^{k} \frac{(1+\alpha) \ldots(1+(k-1) \alpha)}{k!} p_{0}\) for \(k>0\) where \(\alpha>0, \lambda>0\) and
\[ p_{0}=P\{\xi=0\}=(1+\alpha \lambda)^{-\frac{1}{\alpha}} \]

Transcribed Image Text:

Example 5. The random variable (, n) is distributed according to the normal law: p(x, y)= 2 1 2r exp{-21-2+ ] } xy Find the distribution function of the quotient (= 1.