Probability Generating Function of Shifted Random Variable

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $X$ be a discrete random variable whose probability generating function is $\map {\Pi_X} s$.

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

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


Then

$\map {\Pi_Y} s = s^m \map {\Pi_X} s$

where $\map {\Pi_Y} s$ is the probability generating function of $Y$.


Proof

From the definition of p.g.f:

$\ds \map {\Pi_X} s = \sum_{k \mathop \ge 0} \map \Pr {X = k} s^k$

By hypothesis:

$\map \Pr {Y = k + m} = \map \Pr {X = k}$

Thus:

\(\ds \map {\Pi_Y} s\) \(=\) \(\ds \sum_{k + m \mathop \ge 0} \map \Pr {X = k} s^{k + m}\)
\(\ds \) \(=\) \(\ds \sum_{k + m \mathop \ge 0} \map \Pr {X = k} s^m s^k\)
\(\ds \) \(=\) \(\ds s^m \sum_{k \mathop \ge 0} \map \Pr {X = k} s^k\) Translation of Index Variable of Summation


This needs considerable tedious hard slog to complete it.
In particular: The last step needs to be expanded and explained as to why we can just lose the $m$ out of the index.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Finish}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


From the definition of a probability generating function:

$\map {\Pi_Y} s = s^m \map {\Pi_X} s$

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Probability_Generating_Function_of_Shifted_Random_Variable&oldid=525729"