antennense

2022-07-14

Given the following probability mass function, determine C.
Given the following probability mass function:
${P}_{xy}\left(x,y\right)=C{\left(\frac{1}{2}\right)}^{x}\cdot {\left(\frac{1}{2}\right)}^{y}$ determine C.
Hint: use the definition for a geometric series :
$\sum _{n=0}^{\mathrm{\infty }}{r}^{n}=\frac{1}{\left(1-r\right)}$
I'm not quite sure what to even do, I would normally integrate to find a constant? but I'm using geometric series now, and I don't really have any values for x,y or ${P}_{xy}\left(x,y\right)$...<br<x and y can take any integer value equal or bigger then 0

vrtuljakc6

Expert

Explanation:
Because $\sum _{\left\{x,y\right\}=0}^{\mathrm{\infty }}{\left(\frac{1}{2}\right)}^{x}{\left(\frac{1}{2}\right)}^{y}=\frac{1}{1-1/2}\frac{1}{1-1/2}=4$, the normalization constant must be $C=1/4$

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