Explain why the function [latex]F(z)=z^2+z-1[/latex] is or is not a valid z-transform of the PMF of a random variable.

One of the tests for a function of z to be a valid z-transform of a PMF is that it must be equal to 1 when evaluated at [latex]z=1[/latex]. As can be seen, [latex]F(1)=1[/latex], so the function has passed the first test. The second test is that the coefficients of [latex]z[/latex] must be nonnegative since, for example, the coefficient of [latex]z^k[/latex] is the probability that the random variable takes the value [latex]k[/latex]. In the function above, the constant term, which represents the probability that the supposed random variable takes the value 0, is negative 1. This means that the function cannot be a valid z-transform of a PMF.

Question 7.6: Explain why the function F(z) = z² + z - 1 is or is not a va......

Fundamentals of Applied Probability and Random Processes [1392375]

Explain why the function $F(z)=z^2+z-1$ is or is not a valid z-transform of the PMF of a random variable.

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One of the tests for a function of z to be a valid z-transform of a PMF is that it must be equal to 1 when evaluated at $z=1$ . As can be seen, $F(1)=1$ , so the function has passed the first test. The second test is that the coefficients of $z$ must be nonnegative since, for example, the coefficient of $z^k$ is the probability that the random variable takes the value $k$ . In the function above, the constant term, which represents the probability that the supposed random variable takes the value 0, is negative 1. This means that the function cannot be a valid z-transform of a PMF.