Generating function(really simple????)

Question

With $X_n$ as a discrete random variable that takes the values $1,2,\dots,n$ with equal probability $\frac1n$.

I want to:

Evaluate the probability generating function of $X_n$!!!

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$G_{X_n}= \sum_{x=1}^n P(X_n = x) z^x = \sum_{x=1}^n \frac1n z^x=z^x$

Is it really that simple?!?

Answer 1

Geometric series: from first principles

Let: $$S_n = \sum\limits_{x=1}^n z^x = z + z^2 + \cdots + z^{n-1} + z^n$$

Then: $$ z\cdot S_n = \sum\limits_{x=1}^n z^{x+1} = z^2 + \cdots + z^n + z^{n+1}$$

Telescoping: $$(1-z)S_n = z - z^{n+1}$$

Finally: $$S_n = \dfrac{z(1-z^n)}{1-z}$$

Answer 2

It should be \begin{align} G_{X_n}(z) &= \sum_{1 \le x \le n} \frac{z^x}{n} \\ &= \frac{z(1 - z^n)}{n (1 - z)} \end{align}