Question about continuous and discrete moment generating functions.

[The1TL]

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?

 

[Ray Vickson]

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?

There is a theorem that two cdfs are the same if and only if their characteristic functions are the same; in other words, a characteristic function can belong to only one cdf. Of course, the mgf is the characteristic function at an imaginary argument, so the result still applies. Just Google "characteristic function" for more details.