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This is surely a very simple, well-known fact about moment-generating functions, though I am interested in the rigor required to prove it.

It is surely the case that for any random variable, $X$, which could be either discrete or continuous, we have $M_X (0) = 1$. I can think of two possible ways to prove this. The first way surely will always work: we set up either a sum or an integral to compute $E(e^{tx})$. The second way seems plausible, but I do not know whether it is valid: \begin{align*} M_X (t) = E(e^{0 \cdot X}) = E(e^0) = E(1) = 1. \end{align*} What I worry about, in particular, is the second equality. $0$ is surely a constant, $X$ a random variable. The product should be another random variable, I assume -- or is it? It usualy is the case that scaling a random variable by a constant returns another random variable, but multiplying by $0$ removes any "randomness." Does this simply give us a constant?

I would greatly appreciate any helpful thoughts on this.

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    $\begingroup$ Yes, multiplying a random variable by $0$ makes it non-random. $\endgroup$ – Kavi Rama Murthy Apr 3 at 6:41

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