List questions from moment-generating-functions

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A proof regarding the moment generating function.

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Showing inequality: $pe^{x(1-p)}+(1-p)e^{-xp} \leq e^{x^2(3/4)p}$ for $0 \leq p \leq 1/2, 0 \leq x \leq 1$?

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Proof verification : Inequality involving moment generating function (MGF)

Confidence Interval for Chi-squared distribution

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Stochastic Geometry : Obtaining an Integral

Showing for a variable $X$ with mean and variance 1 that $\mathbb{E}\left(e^{X}\right)\geq e$

A proof regarding the moment generating function.

Is the sample mean of a Poisson Distribution Normal?

Determine pdf from mgf

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Proof of the Central Limit Theorem using moment generating functions

Suggestion for probability problem book

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