Suppose we have a continuous random variable $X$. We have a function of this $X$ given as follows: $f(X)=K_1 X - K_3 X^2$ where $K_1$, $K_3$ are constants. We also have a small, positive constant $k$. Does it make sense to do a marginal analysis on $X$, i.e. can we write $$\begin{align}E[f(X+k)-f(X)] &= E[(K_1 (X+k) - K_3 (X+k)^2)-(K_1 X - K_3 X^2)]\\ &= K_1 k-K_3 k^2 -2 k K_3 E[X]?\end{align}$$ Is this technically correct?
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2"Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking." – Did Jul 19 '13 at 10:51
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edited with more clarification. – Pradipta Jul 19 '13 at 12:02
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Now the question is clear--and seems entirely trivial: yes, $E(f(X+k)-f(X))$ is what you say (but I have no idea why you call this a marginal analysis on $X$). – Did Jul 19 '13 at 19:57
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Actually the purpose was to see that if you increase $X$ by a small quantity $k$ then how does $E(f(X+k))$ change? – Pradipta Jul 20 '13 at 05:27