Question: Let X have the Poisson distribution with mean in range of (0;inf). What does Jensen's inequality say about E[exp(-X)]? What is the actual value of

Let X have the Poisson distribution with mean in range of (0;inf). What does Jensen's inequality say about E[exp(-X)]? What is the actual value of E[exp(-X)]?

Hint: You may use, without proof, the formula for the moment generating function of a Poisson distribution.

Jensen's Inequality: Let X be a random variable, and suppose that g is convex. (We may restrict the domain of g, if necessary, to a convenient interval that contains the support of X.) If E[X]

exists finitely, then E[g(X)] g(E[X])

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