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I am doing the following exercise 5.7.11 from Grimmett & Stirzaker - Probability and Random Processes.

I have done the second part about the moment generating function.

However, I don't see why $F_\tau$ is a distribution function. I know it has to be nondecreasing, right-continuous and the limits must be 0 and 1. I have the solution from the manual but I don't understand it. Could someone provide any help?

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