Find the generating functions, both ordinary h(z) and moment g(t), for the following discrete probability distributions.
Note: \(h(z) = g(log z)\) and \(g(t) = h(e^{t})\). This can be found on page 369.
\(p_{X} = \begin{pmatrix} \text{Heads} & \text{Tails} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}\)
\(h(z) = 0z^{0}+\frac{1}{2}z^{1}+\frac{1}{2}z^{2}\)
\(h(z) = 0 + \frac{1}{2}z(1+z)\)
\(h(z) = \frac{1}{2}z(1+z)\)
The ordinary generating function for a distribution describing a fair coin is \(\frac{1}{2}z(1+z)\).
\(g(t) = h(e^{t})\)
\(g(t) = \frac{1}{2}(1+e^{t})\)
The moment generating function for a distribution describing a fair coin is \(\frac{1}{2}e^{t}(1+e^{t})\).
\(p_{X} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} & \frac{1}{6} \end{pmatrix}\)
\(h(z) = 0z^{0}+\frac{1}{6}z^{1}+\frac{1}{6}z^{2}+\frac{1}{6}z^{3}+\frac{1}{6}z^{4}+\frac{1}{6}z^{5}+\frac{1}{6}z^{6}\)
\(h(z) = 0 + \frac{1}{6}z(1+z+z^{2}+z^{3}+z^{4}+z^{5})\)
\(h(z) = \frac{1}{6}z(1+z+z^{2})(z^{3}+z^{4}+z^{5})\)
\(h(z) = \frac{1}{6}z(1+z+z^{2})+z^{3}(1+z+z^{2})\)
\(h(z) = \frac{1}{6}z(1+z+z^{2})(z^{3}+1)\)
\(h(z) = \frac{1}{6}z(1+z+z^{2})(z+1)(1-z+z^{2})\)
The ordinary generating function for a distribution describing a fair die is \(\frac{1}{6}z(1+z+z^{2})(z+1)(1-z+z^{2})\). More succinctly, it’s \(\sum\limits^{6}\limits_{j=1}z^{j}\)
\(g(t) = h(e^{t})\)
\(g(t) = \frac{1}{6}e^{t}(1+e^{t}+e^{2t})(e^{t}+1)(1-e^{y}+e^{2t})\)
The moment generating function for a distribution describing a fair die is \(\frac{1}{6}e^{t}(1+e^{t}+e^{2t})(e^{t}+1)(1-e^{y}+e^{2t})\). More succinctly, it’s \(\sum\limits^{6}\limits_{j=1}e^{jt}\).
\(p_{X} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 0 & 0 & 1 & 0 & 0 & 0 \end{pmatrix}\)
\(h(z) = 0z^{0}+0z^{1}+0z^{2}+1z^{3}+0z^{4}+0z^{5}+0z^{6}\)
\(h(z) = 0 + 0 + 0 + z^{3} + 0 + 0 + 0\)
\(h(z) = z^{3}\)
The ordinary generating function for a distribution describing a die that always comes up 3 is \(z^{3}\).
\(g(t) = h(e^{t})\)
\(g(t) = e^{3t}\)
The moment generating function for a distribution describing a die that always comes up 3 is \(e^{3t}\).
\(h(z) = 0z^{0} + \frac{1}{k+1}z^{n} + \frac{1}{k+1}z^{n+1} + \frac{1}{k+1}z^{n+2} + ... + \frac{1}{k+1}z^{n+k}\)
\(h(z) = 0 + \frac{1}{k+1}(z^{n}+z^{n+1}+z^{n+2}+...+z^{n+k})\)
\(h(z) = 0 + \frac{1}{k+1}z^{n}(1+z^{n}+z^{n+1}+...+z^{n+k-1})\)
\(h(z) = \frac{1}{k+1}z^{n}\sum\limits^{n+k}\limits_{j=1}z^{j}\)
The ordinary generating function for the uniform distribution on the set \({n, n+1, n+2, ..., n+k}\) is \(\frac{1}{k+1}z^{n}\sum\limits^{n+k}\limits_{j=1}z^{j}\).
\(g(t) = h(e^{t})\)
\(g(t) = \frac{1}{k+1}e^{nt}\sum\limits^{n+k}\limits_{j=1}e^{jt}\)
The moment generating function for the uniform distribution on the set \({n, n+1, n+2, ..., n+k}\) is \(\frac{1}{k+1}e^{nt}\sum\limits^{n+k}\limits_{j=1}e^{jt}\).
\(h(z) = \sum\limits^{n}\limits_{j=0}z^{j}\begin{pmatrix}n \\ j\end{pmatrix}p^{j}q^{n-j}\)
\(h(z) = \sum\limits^{n+k}\limits_{j=n}z^{n}\begin{pmatrix}n+k \\ n\end{pmatrix}p^{n}q^{n+k-n}\)
\(h(z) = \sum\limits^{n+k}\limits_{j=n}z^{n}\begin{pmatrix}n+k \\ n\end{pmatrix}p^{n}q^{k}\)
\(h(z) = z^{n}(pz+q)^{k}\)
The ordinary generating function for the binomial distribution on the set \({n, n+1, n+2, ..., n+k}\) is \(z^{n}(pz+q)^{k}\).
\(g(t) = h(e^{t})\)
\(g(t) = e^{nt}(pe^{t}+q)^{k}\)
The moment generating function for the binomial distribution on the set \({n, n+1, n+2, ..., n+k}\) is \(e^{nt}(pe^{t}+q)^{k}\).
\(h(z) = \sum\limits^{\infty}\limits_{j=0}z^{j}\frac{2}{3}^{j+1}\)
\(h(z) = \frac{2}{3-z}\)
The ordinary generating function for the geometric distribution on the set \({0, 1, 2, ...,}\) with \(p(j) = 2/3^{j+1}\) is \(\frac{2}{3-z}\).
\(g(t) = h(e^{t})\)
\(g(t) = \frac{2}{3-e^{t}}\)
The ordinary generating function for the geometric distribution on the set \({0, 1, 2, ...,}\) with \(p(j) = 2/3^{j+1}\) is \(\frac{2}{3-e^{t}}\).