Continuous differentiability of $\log({\bf{E}}\exp{\lambda X})$ in a given fixed domain

Question

What is the proof that, for a R.V $X$, $\log\big({\bf{E}}[\exp(\lambda X)]\big)$ is continuously differentiable w.r.to $\lambda \in [0,a)$.

Answer 1

As suggested by Gabriel Romon, you should look at when a function can be differentiated under the integral (or expectation) sign. He gives a valid reference, but personally I'd recommend looking at these lecture notes (on probability and measure theory), specifically Section 3.5 (page 24 onwards), titled "Differentiation Under the Integral Sign"

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Here's a method that will work, but isn't such a great idea, as is referenced in the comments. I saw this just before having to go, and this was the first idea that popped into my head!

I won't give all the details, but instead give an overall idea. I leave the details to you as an exercise. (I feel one learns much more by doing maths rather than being told maths.)

Note that $\log$ is a smooth function, and so we may as well ignore it. I assume that $a$ is defined so that $E(\exp(\lambda X))$ exists for all $\lambda \in [0,a)$? If so, my recommendation would be to write the exponential as a (Taylor) series, for $\lambda$ inside this interval. You can then use results like Fubini to move the expectation through the infinite sum. This gives a standard infinite sum (since once you take the expectation you get a standard real). You then have results from real analysis about when infinite sums; apply these. (If you've done a course with a name like Analysis I, then you'll have (likely) covered these results there.)