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The original question is that $X$ is a random variable that $E(X)=\mu$. We are interested in statistical functionals $\theta=\int\log(x)dF(x)$ and $ \lambda=\log(\mu)$.

The first part of the question asks us to derive the influence function for $\theta$ which is simple, provided that $\theta$ is a linear functional. For the second part of the question, we are asked to derive the influence function for $\log \mu$ which puzzles me because $\mu$ is log of the expected value of $X$ which is linear. Could someone please show me how to derive the influence function in this case or gives me some hints?

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