Compute the moment generating function of $X$ [closed]

Suppose that $X$ has density function $$f(x) = e^{−x}, \quad x > 0.$$ Compute the moment generating function of $X$ and use your result to determine its mean and variance. Check your answer for the mean by a direct calculation

My attempt: \begin{align*} f(x) & = e^{-x} \\ M(t) & = \int_0^\infty e^{tx}f(x) ~dx \\ & = \int_0^\infty e^{tx}e^{-x} ~dx \\ & = \int_0^\infty e^{x(t - 1)} ~dx \\ & = \left. \frac{e^{x(t - 1)}}{t - 1} \right|_{t = 0}^\infty \\ & = \frac{1}{1 - t}, \text{ for } t < 1. \\ M'(t) & = \frac{1}{1 - t}. \\ E[X] & = M'(0) = 1. \\ M''(t) & = \frac{2}{(1 - t)^3}. \\ E[X^2] & = M''(0) = 2. \\ \mathrm{Var}(X) & = E[X^2] - (E[X])^2 = 2 - 1^2 = 1. \\ \end{align*} To check those answers, evaluate these integrals: \begin{align*} E[X] & = \int_0^\infty xe^{-x} ~dx = 1. \\ E[X^2] & = \int_0^\infty x^2 e^{-x} ~dx = 2. \end{align*}

Is it right to my procedure?

-
I don't see a question here. –  Qiaochu Yuan May 21 '12 at 0:37
If your question is whether the mgf is $1/(1-t)$, the answer is that it is. For finding $E(X)$, $E(X^2)$ by direct integration, use integration by parts. –  André Nicolas May 21 '12 at 0:43
Is it right to my procedure? –  Daniela del Carmen May 21 '12 at 0:46
@Henry T. Horton: Nice job of making the post look good. –  André Nicolas May 21 '12 at 0:47
@Daniela del Carmen: Your mgf calculation is fine (as interpreted by Henry T. Horton), also the calculation of mean, variance from the mgf. To answer the question fully, you are being asked also to find the two expectations directly. –  André Nicolas May 21 '12 at 0:50