Moment generation function for continuous random variable

Question

Apologies, my math skills have become rusty and I'm having trouble with this question:

$X$ is a continuous random variable with $$f(x)=\cases{e^{-(x+2)}&\text{if } -2 < x < \infty\\0&\text{otherwise}}.$$

Find the MGF of $X$.

So far I have started with this below, but I'm unsure of how to find the derivative from here. I've only seen problems where $x$ is between $0$ and $\infty$ before so I'm not sure how to deal with $-2$.

$$M_X(t)=\int_{-2}^{\infty}e^{tx}e^{−(x+2)} dx$$

Thanks very much.

Answer 1

Hint: $$e^{tx}e^{-(x+2)} = e^{(t-1)x-2}=\frac{d}{dx}\left(\frac{e^{(t-1)x-2}}{t-1}\right)$$ Now integrate both sides with respect to $x$ over $[2,\infty)$.