Answer:
Step-by-step explanation: To find the moment-generating function (MGF) for the random variable Y, which is defined as Y = -2X + 7, we can use the properties of MGFs.
1. The MGF of a linear transformation of a random variable X is given by:
M_Y(t) = E[e^(tY)] = E[e^(t(-2X+7))] = E[e^(-2tX+7t)]
2. Since X follows a binomial distribution with n=12 trials and p=0.35, the MGF of X is:
M_X(t) = E[e^(tX)] = (0.35e^t + 0.65)^12
3. Now, substitute the MGF of X into the expression for M_Y(t):
M_Y(t) = E[e^(-2tX+7t)] = E[e^(-2tX)e^(7t)]
4. Using the property that the MGF of the sum of independent random variables is the product of their MGFs, we get:
M_Y(t) = M_X(-2t) e^(7t) = (0.35e^(-2t) + 0.65)^12 e^(7t)
5. Therefore, the moment-generating function for Y, denoted as M_Y(t), is:
M_Y(t) = (0.35e^(-2t) + 0.65)^12 * e^(7t)
The correct option related to the MGF for Y among the given choices is not explicitly mentioned in the question. However, the accurate expression derived above matches closest to the provided options.
