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Question 7.5: Is the following function a valid z-transform of a PMF? If s......

Is the following function a valid z-transform of a PMF? If so, what is the PMF?

\,\displaystyle g\left (z\right )=\frac{1-a}{1-a z} \quad 0<a<1
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First, we note that g(1)=1 , which means that the function g(z) is potentially a valid z-transform of a PMF. Next, we test the coefficients of z in the function. Now,

\,\displaystyle g\left (z\right )=\left (1-a\right ) \sum_{k=0}^{\infty}\left (a z\right )^k=\left (1-a\right )\left\{1+a z+a^2 z^2+a^3 z^3+\cdots+a^k z^k+\cdots\right\}

Since 0 \lt a \lt 1, we see that all the coefficients of z are nonnegative quantities that are no greater than 1. Therefore, the function is a valid z-transform of a PMF. If X is the random variable whose PMF has this z-transform, the PMF of X is given by

\,\displaystyle p_X\left (x\right )=\left (1-a\right ) a^x \quad x=0,1,2, \ldots

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