1. **Problem Statement:** Find the moment generating function (MGF) $M_X(s)$ and all moments $E[X^k]$ of a random variable $X$ with exponential distribution parameter $\lambda$, where $0 \leq x < \infty$ and PDF $f_X(x) = \lambda e^{-\lambda x} d(x)$, with $d(x)$ the unit step function. 2. **MGF Definition:** The MGF of $X$ is defined as $$M_X(s) = E[e^{sX}] = \int_0^\infty e^{sx} f_X(x) \, dx$$ 3. **Calculate MGF:** Substitute $f_X(x)$: $$M_X(s) = \int_0^\infty e^{sx} \lambda e^{-\lambda x} \, dx = \lambda \int_0^\infty e^{(s - \lambda)x} \, dx$$ 4. **Convergence Condition:** For the integral to converge, we need $s - \lambda < 0 \Rightarrow s < \lambda$. 5. **Evaluate Integral:** $$\int_0^\infty e^{(s - \lambda)x} \, dx = \left[ \frac{e^{(s - \lambda)x}}{s - \lambda} \right]_0^\infty = \frac{1}{\lambda - s}$$ 6. **Final MGF:** $$M_X(s) = \frac{\lambda}{\lambda - s}, \quad s < \lambda$$ 7. **Moments from MGF:** The $k$-th moment is given by $$E[X^k] = M_X^{(k)}(0) = \left. \frac{d^k}{ds^k} M_X(s) \right|_{s=0}$$ 8. **Derivatives:** Since $$M_X(s) = \frac{\lambda}{\lambda - s} = (1 - \frac{s}{\lambda})^{-1}$$ Using the binomial series for negative powers, $$M_X(s) = \sum_{k=0}^\infty \frac{k!}{\lambda^k} s^k$$ 9. **Explicit Moments:** Therefore, $$E[X^k] = \frac{k!}{\lambda^k}$$ **Summary:** - MGF: $$M_X(s) = \frac{\lambda}{\lambda - s}, \quad s < \lambda$$ - Moments: $$E[X^k] = \frac{k!}{\lambda^k}$$