Let $p$ be a probability distribution on $(0,1,2)$ with moments $\mu_1=1$ and $\mu_2=\frac{3}{2}$
Find its ordinary function $h(z)$. Using this then find its Moment generating function. Then find its first 6 moments. Finally find $p_0 , p_1, p_2$.
I am trying to solve the above problem but all of the examples I have worked through in my class provided more information such as providing a formula for $u_k$ which this question does not. I get the difference between the two generating functions but I do not know how to start this.
Let $X$ have distribution $p$. From $\mathbb E[X]=1$ and $\mathbb E[X^2]=\frac32$ we have \begin{align} p_1 + 2p_2 &= 1\\ p_1 + 4p_2 &= \frac32, \end{align} and hence $p_1=\frac12$, $p_2=\frac14$. Since $p_0+p_1+p_2=1$, it follows that $p_0=\frac14$. The generating function $h(z):=\mathbb E\left[z^X\right]$ of $X$ is thus $$ h(z) = \frac14 + \frac12 z + \frac14 z^2 =\frac14(1+z)^2, $$ and the moment generating function $$ \mathbb E\left[e^{tX}\right] = \frac14 + \frac12 e^t + \frac14 e^{2t} = \frac14(1+e^t)^2. $$