Consider a random variable V modeling the version of the software that people have installed on their computers, where the possible versions are {1, 2, 3, 4}. People are equally likely to have versions 1 and 2 and also equally likely to have versions 3 and 4. People are also x times more likely to have version 3 than version 2. (a) What is the (probability mass function) PMF of V as a function of x (b) What is the expected version of the software. Hint: the expected value of V as a function of x (c) What is the minimum value of x to ensure that the mean version is at least version 3?

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