Consider X∼Geometric(p)X \sim \text{Geometric}(p)X∼Geometric(p). Find the MGF MX(t)=E[etX]M_X(t) = E[e^{tX}]MX(t)=E[etX] for t<−ln(1−p)t < -\ln(1-p)t<−ln(1−p).
pet1−(1−p)et\frac{pe^t}{1-(1-p)e^t}1−(1−p)etpet
p1−(1−p)et\frac{p}{1-(1-p)e^t}1−(1−p)etp
pet1−pet\frac{p e^t}{1-pe^t}1−petpet
pet+(1−p)pe^t + (1-p)pet+(1−p)