The memoryless property characterizes a distribution by the condition P(X > s+t | X > s) = P(X > t), and the exponential distribution is the unique continuous probability distribution possessing this property, proven by reducing the memoryless condition to a functional equation G(s+t) = G(s)G(t) for the survival function and solving it via successive extension from integers to rationals to reals under continuity, yielding an exponential form. The moment generating function (MGF), defined as M(t) = E(e^{tX}), is an alternative characterization of a probability distribution whose Taylor expansion encodes all moments of the distribution, uniquely determines the distribution, and converts the distribution of a sum of independent random variables into a simple product of MGFs, making it a central tool in probability theory for analyzing distributions and sums of random variables.
The memoryless property characterizes a distribution by the condition P(X > s+t | X > s) = P(X > t), and the exponential distribution is the unique continuous probability distribution possessing this…