Final answer:
To find the moment generating function, mean, and variance for the given PMF, we can use the formula MGF(t) = E(e^tx), where E denotes the expectation. The mean is the first derivative of the MGF evaluated at t = 0, and the variance is the second derivative of the MGF evaluated at t = 0 minus the square of the mean.
Step-by-step explanation:
The given probability mass function (PMF) is f(x) = 1/2(2/6)x for x = 1, 2, 3, 4, ... To find the moment generating function (MGF), we can use the formula MGF(t) = E(etx), where E denotes the expectation. We need to calculate the expectation of etx for each x value in the PMF and then sum them up. This will be the MGF for the given PMF. The mean is the first derivative of the MGF evaluated at t = 0. The variance is the second derivative of the MGF evaluated at t = 0 minus the square of the mean.