Thank you for your messages, indicating that there is a mistake in the question. Please find the moment generating function for x=1, 2, and then please solve the amended question:
Given f(x)=(1/2)^x, for x=0,1,2,…, and zero elsewhere, please show that its moment generating function is M(z)=2(2-e^z)^-1. Please use it to calculate the meanand varianceof X.
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