Probability Generating Function of Shifted Random Variable

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Theorem

Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$.

Let $k \in \Z_{\ge 0}$ be a positive integer.

Let $Y$ be a discrete random variable such that $Y = X + m$.


Then

$\Pi_Y \left({s}\right) = s^m \Pi_X \left({s}\right)$.

where $\Pi_Y \left({s}\right)$ is the probability generating function of $Y$.


Proof

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{k \mathop \ge 0} \Pr \left({X = k}\right) s^k$

By hypothesis:

$\Pr \left({Y = k + m}\right) = \Pr \left({X = k}\right)$

Thus:

$\displaystyle \Pi_Y \left({s}\right) = \sum_{k + m \mathop \ge 0} \Pr \left({X = k}\right) s^{k + m}$
\(\displaystyle \Pi_Y \left({s}\right)\) \(=\) \(\displaystyle \sum_{k + m \mathop \ge 0} \Pr \left({X = k}\right) s^{k + m}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k + m \mathop \ge 0} \Pr \left({X = k}\right) s^m s^k\)
\(\displaystyle \) \(=\) \(\displaystyle s^m \sum_{k \mathop \ge 0} \Pr \left({X = k}\right) s^k\) Translation of Index Variable of Summation



From the definition of a probability generating function:

$\Pi_Y \left({s}\right) = s^m \Pi_X \left({s}\right)$

$\blacksquare$


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