Final answer:
To find the PMF of Y, differentiate the moment-generating function and evaluate it at t = 0. Use the moment-generating function to find E(Y) by taking the first derivative at t = 0. Use the PMF of Y to determine E(Y) by taking the weighted average of the possible outcomes. Both methods give the same result for E(Y).
Step-by-step explanation:
To find the probability mass function (PMF) of the random variable Y, we need to find the values of the probabilities for each possible outcome of Y. The moment-generating function m(t) can be used to find the PMF by differentiating it. In this case, m(t) = 0.1e^t + 0.2e^(2t) + 0.3e^(3t) + 0.4e^(4t).
To find E(Y) using the moment-generating function, we can take the first derivative of the moment-generating function and evaluate it at t = 0. The derivative of m(t) with respect to t gives us the moment-generating function for Y. From the moment-generating function, we can find the expected value E(Y) by taking the first derivative at t = 0. In this case, E(Y) = m'(0) = 0.1 + 0.4.
Using the PMF of Y, we can determine E(Y) by taking the weighted average of the possible outcomes of Y, where each outcome is multiplied by its corresponding probability. The expected value E(Y) is given by the sum of all possible outcomes of Y multiplied by their probabilities. In this case, E(Y) = 1 * 0.1 + 2 * 0.2 + 3 * 0.3 + 4 * 0.4 = 2.8.
Comparing the results from part b and part c, we can see that both methods give us the same value for the expected value E(Y), which is 2.8.