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I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook:

Suppose X is a random variable for which the pdf is:

$$f(x) = \begin{cases}e^{-x} & x > 0\\0 & otherwise\end{cases} $$

Determine the mgf of X.

$$\psi(t) = E(e^{tX}) \int_0^\infty e^{tx}e^{-x}dx $$ $$ = \int_0^\infty e^{(t-1)x}dx $$

I understand the above, but here is where I am confused. The book then says: "The final integral in this equation will be finite if and only if $t < 1$. Therefore, $\psi(t)$ is finite only for $t<1$."

1) How do you determine that it's only finite for $t<1$?

Additionally, it goes on to solve the integral: $\psi(t) = \frac{1}{1-t}$, then it states: "Since $\psi(t)$ is finite for all values of $t$ in an open interval around the point $t=0$, all moments of X exist."

2) What does it mean by an open interval around $t=0$?

Please help me understand. Thanks in advance!

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  • $\begingroup$ If the textbook does not have an appendix defining terms like "open interval," then it likely has an introduction laying out the prerequisites for understanding it--and they would include a course in basic topology or a reasonably rigorous course in analysis, such as an intro Calculus course given at a top-tier university or a capstone analysis course for math majors at other institutions. That gives you a guide to researching materials for learning these prerequisites. $\endgroup$
    – whuber
    Dec 3, 2019 at 16:43
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    $\begingroup$ @whuber - This is actually covered in high school AP calculus these days... at least the one my daughter is taking. $\endgroup$
    – jbowman
    Dec 3, 2019 at 17:32
  • $\begingroup$ Please add the self-study tag and read its wiki. $\endgroup$ Dec 3, 2019 at 18:07

1 Answer 1

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Hint: The integrand $e^{(t-1)x}$ has value $1$ if $t=1$ and is an increasing function of $x$ if $t > 1$. What do you suppose is the value of the integral when $t \geq 1$? In particular, is the value a finite number?

For the definition of open interval, do some searching for yourself.

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