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The moment generating function (mgf) is an incredibly useful tool in probability theory, as it uniquely characterizes a probability distribution and simplifies the process of finding moments (like the mean and variance). An mgf is defined for a random variable X as the expected value of the exponential function of tx, where t is a real number, expressed as M_X(t) = E[exp(tX)].

When we say it 'generates' moments, this refers to the fact that by taking derivatives of the mgf and then evaluating at t=0, we get the moments of the distribution. For instance, the first derivative gives us the first moment (the mean), the second derivative gives us the second moment (which is related to the variance), and so on.

For the exercise at hand, the joint mgf for three random variables, X1, X2, and X3, was computed. The integral involved exp[s1x1 + s2x2 + s3x3 - (x1 + x2 + x3)], which led to a product of terms, each in the form of 1/(1 - si) for si > -1. This result is crucial because it gives an insight into the relationship between the variables.

A probability distribution is a mathematical description of the relative likelihood of different outcomes. It quantifies the probabilities associated with the possible values of a random variable. Continuous random variables, such as the ones in our exercise X1, X2, and X3, have a probability distribution described by a probability density function (pdf).

The joint pdf f(x1, x2, x3) given in the exercise represents the distribution of three variables together. From this, probabilities of events that involve relationships between these variables are calculated by integrating the pdf over the region of interest. The exercise asked for the probability of the event X1 < X2 < X3, which required integrating the joint pdf over the region where x1, x2, and x3 satisfy that inequality, resulting in the probability of 1/6.

Independence of random variables is a key concept in probability and statistics. Two or more random variables are independent if the occurrence of one does not affect the probabilities associated with the others. In mathematical terms, for two variables X and Y, they are independent if for all x and y, P(X=x and Y=y) = P(X=x)P(Y=y).

For the joint mgf to indicate independence, it should factorize into a product of the individual mgfs of the random variables. In our exercise though, when the joint mgf does not simplify into the product of individual mgfs, it implies that the variables are not independent. This was observed in the final step of the solution, where the joint mgf of X1, X2, and X3 couldn't be separated into individual factors, hence demonstrating that they are dependent on each other.