If a random variable X assumes the value of a Bernoulli(1/3) random variable with a probability of 1/4, and assumes the value of a Uniform(0,1) random variable with a probability of 3/4, how does one obtain a moment generating function of X?
I'm fuzzy on how exactly mixed distributions work, and I'm not sure that term even applies here, however I came across this problem and thought it to be like no other.
In any case, I believe the moment generating function for the uniform part of this variable would be ((e^t)-1)/t. And the Bernoulli mgf would be 2/3+(1/3)(e^t).
Given that the Bernoulli variable occurs with probability 3/4 and the Uniform variable with probability 1/4, how does one combine these into one cohesive mgf? Have I simply forgotten a formula?