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Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate

$$\frac{\partial}{\partial \alpha} \mathbb{E} [f(x)]$$

Ultimately, this is for the following purpose: Suppose $\alpha^*(y)$ is defined by the equation

$$\mathbb{E} [f(x) \mid \alpha] \equiv \alpha + y$$

I am interested in studying the implicit function $\alpha^*(y)$ so defined. Any tips to characterize it would be greatly appreciated.

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+1. I changed \;|\; to \mid. I think that is standard. –  Michael Hardy Oct 4 '13 at 21:19
    
So $\alpha^*(y)$ is the value of $\alpha$ for which the bias of $f(x)$ as an estimator of $\alpha$ is $y$. –  Michael Hardy Oct 4 '13 at 21:20
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1 Answer

What does the notation $\mathbb{E}[ \cdot \mid \alpha]$ mean?

Please tell us more about what this problem is for, but the paper referenced at expected values over binomial distributions by Sah ("The Effects of Child Mortality Changes on Fertility Choice and Parental Welfare", equations 4 - 7) offers a property which sufficed for a similar problem I was tackling. Namely,

\begin{align*} \frac{\partial}{\partial \alpha} \mathbb{E}[f(x)] = k \cdot \sum_{j=0}^{k-1} {k-1 \choose j} \alpha^j (1-\alpha)^{k-1-j} (f(x+1)-f(x)). \end{align*}

Lots of other nice properties are derived in this paper, and the answers to that question offer some further reading.

Edit: See also Is the Binomial Expectation of Convex Function Convex in p? which has much more analysis of this type.

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The notation $\mathbb E[f(x)\mid\alpha]$ is a way of writing the expected value in which its dependence on $\alpha$ is given a place in the notation, so that, for example, one can write expressions like $\mathbb E[f(x)\mid\alpha]/\mathbb E[f(x)\mid\alpha_0]$. Suppose that 20 pages after you first write about the random variable $x$ whose distribution depends on $\alpha$, it becomes appropriate to assign a probability distribution to $\alpha$. Then the notation refers to the conditional expected value while remaining consistent with the way it was used earlier. –  Michael Hardy Oct 4 '13 at 21:16
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