Final answer:
M_X(t) = t / (1 - t) cannot be the MGF of a random variable because it does not remain finite for all t in an interval around zero; as t approaches 1, the function becomes infinite, violating the conditions required for moment-generating functions.
Step-by-step explanation:
The question pertains to the properties of moment-generating functions (MGFs) of random variables. A moment-generating function, M_X(t), is a function that summarizes all of the moments of a random variable. For a moment-generating function to be valid, it must satisfy certain properties, one of which is that it must exist (be finite) for all values of t in some open interval containing zero.
In the case of M_X(t) = t / (1 - t), this function is not valid for a moment-generating function because it does not satisfy the necessary property of being finite for all values of t near zero. As t approaches 1, the denominator (1 - t) approaches zero, and hence M_X(t) approaches infinity, which invalidates it as a moment-generating function for a random variable.
To explore this further, the exponential expansion of M_X(t) would lead to terms with increasing powers of t, which implies moments of all orders, some of which would be infinite. This contradicts the fundamental requirement for all moments of a random variable to be finite for the existence of its MGF.