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The probability distribution for count is dist = ParameterMixtureDistribution[ BinomialDistribution[10, p], p \[Distributed] BetaDistribution[1, 1]] (* BetaBinomialDistribution[1, 1, 10] *) The expected value of count is the Mean of dist Mean[dist] (* 5 *) Probability[count > 1/2, count \[Distributed] dist] (* 10/11 *)

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Since in the original code there are instabilities due to low order approximation we can use 4th order numerical algorithm I have developed for the Lotka-McKendrick demographic model (see the very last code in my answer). First we define function f, g using next exact expression for $E(x)$: l0 = -25/10; l1 = 75/10; x0 = 1/2; x1 = 3/2; c = 1; eps = 3/5; L[x_]...

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This is not an answer, but some comments on solving this type of problem that are too long and to made in comments to the question. Regarding scaling up and down: In my opinion, in order to become proficient in solving difficult problems it is imperative to learn how to scale the problem down and then back up again. For example, you have:  \frac{\...

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Clear["Global*"] The integral needs to be done one way or another. You can use Normalize directly f[x_] = Normalize[A Sinc[x], Integrate[#, {x, -Infinity, Infinity}] &] (* Sinc[x]/π *) Verifying, Integrate[f[x], {x, -Infinity, Infinity}] (* 1 *)

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You can exploit the Method -> "Normalize" of ProbabilityDistribution[] for this: PDF[ProbabilityDistribution[a Sinc[u], {u, -∞, ∞}, Assumptions -> Positive[a], Method -> "Normalize"], x] Sinc[x]/π

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You can use ParameterMixtureDistribution Mean @ ParameterMixtureDistribution[PoissonDistribution[λ], Distributed[ λ, NormalDistribution[m, 1]]] Undefined We need to use a distribution with positive support for the distribution of λ: Mean @ ParameterMixtureDistribution[PoissonDistribution[λ], Distributed[ λ, LogNormalDistribution[m, 1]]] E^(1/2 + m) ...

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I posed the question here to a colleague, Vahagn Abgaryan. He responded: "Concerning your question on WishartMatrixDistribution. First, I didn't know that such a function existed, thank you for an interesting finding. Second, this function indeed may be used to generate the real matrices. I include a small program showing the implementation of this ...

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To provide a concrete example Clear["Global*"] Let f[x_] := Log[x] Example 1 distX1 = RayleighDistribution[σ]; m1a = Expectation[f[x], x \[Distributed] distX1] (* -(EulerGamma/2) + Log[2]/2 + Log[σ] *) Alternatively, use TransformedDistribution to define the distribution for f[x] given the distribution for x distF1 = TransformedDistribution[...

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