# Finding the moment generating function of $\min(Y,1)$

Let $$Y\sim\text{Exp}(1)$$ be a random variable. I denote the random variable $$X$$ as $$X=\min(Y,1)$$. The task is to find the moment generating function of $$X$$.

By simply calculating the probability I managed to find the CDF of $$X$$ is: $$F_X(t) = \begin{cases} 1-e^{-t}, & 0 \le t <1\\ 1, & t \ge 1\\ 0, & \text{otherwise} \end{cases}$$

Here I got stuck. Since $$X$$ is not a continuous variable, it does not have a PDF, and without it I do not know how to calculate the moment generating function. ($$X$$ is also not discrete).

I will appreciate some help.

The definition of the moment generating function does not require a PDF. $$\phi_X(t) := E[e^{tX}].$$
From here, it may be helpful to write $$e^{tX} = e^{tX} \mathbf{1}\{Y > 1\} + e^{tX} \mathbf{1}\{Y \le 1\}$$ and compute the expectation of the two terms separately.
$$E[e^{tX} \mathbf{1}\{Y > 1\}] = E[e^{tY} \mathbf{1}\{Y > 1\}] = \int_1^\infty e^{ty} f_Y(y) \, dy = \cdots.$$ $$E[e^{tX} \mathbf{1}\{Y \le 1\}] = e^t E[\mathbf{1}\{Y \le 1\}] = e^t P(Y \le 1) = \cdots.$$