Final answer:
The MGF of a discrete random variable X, such as the number of empty boxes when oranges are distributed, is calculated using the PMF. Without PMF details, exact MGF and the fourth moment E[X^4] cannot be derived. Calculating the expected value involves summing the products of random variable values and their probabilities.
Step-by-step explanation:
The moment-generating function (MGF) for a given random variable X is a tool used in probability theory to characterize the distribution of the variable. In the case where X represents the number of empty boxes when oranges are distributed, the distribution of X is not continuous but discrete since the number of empty boxes can only take integer values. Therefore, the reference to a continuous random variable with a mean μ = 4 minutes and a decay parameter m is not relevant for calculating the MGF of X in this context.
To calculate the MGF, you need to use the probability mass function (PMF) of X, which would give us the probabilities of having 0, 1, 2, 3, or 4 empty boxes. Using this PMF, you can compute the MGF by summing up etx multiplied by the probabilities. Once the MGF is found, the fourth moment, E[X4], can be calculated by differentiating the MGF four times with respect to t and then evaluating it at t=0.
However, without the specific PMF, it is impossible to calculate the precise values for the MGF and the fourth moment E[X4].
To find the expected value E(X), or mean μ, for probability distribution, multiplying each value of the random variable by its probability and adding the products is the standard method.