The discussion centers on the moment generating function (mgf) given by (1/2)(1+e^t) and its implications for continuous and discrete distributions. It is established that this mgf corresponds to a Bernoulli distribution with parameter p = 1/2, which is inherently discrete. Therefore, it is concluded that no continuous random variable can possess this mgf, as the characteristic function theorem indicates that a characteristic function uniquely defines a cumulative distribution function (cdf).
PREREQUISITESStudents and professionals in statistics, probability theory, and data science who are interested in understanding the distinctions between continuous and discrete random variables, particularly in the context of moment generating functions.
The1TL said:Homework Statement
is there a continuous real valued variable X with mgf: (1/2)(1+e^t)
Homework Equations
The Attempt at a Solution
I've noticed that this is the mgf of a bernoulli distribution with p =1/2. But since bernoulli is a discrete distribution, does that disprove that there is a continuous variable with that mgf?