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Suppose I have three discrete-time signals $X_1[n]$, $X_2[n]$ and $Y[n]$ that can be considered as random processes. Suppose they are wide sense stationary. $Y[n]$ is the output of a system $A$. $X_1[n]$ and $X_2[n]$ are the two inputs to the system $A$. $X_1[n]$ and $X_2[n]$ are independent to each other.

The only information I know about system $A$ is that it is causal. That is, $Y[n]$ only depends on $X_1[n-T]$ and $X_2[n-T]$, where $T=0,1,\cdots$. Means of these three random processes are $m_{x_1}$, $m_{x_2}$ and $m_y$, respectively.

Now suppose I have many samples of all these three processes. Is there any way that I can estimate $\frac{\partial m_y}{\partial m_{x_1}}$ and $\frac{\partial m_y}{\partial m_{x_2}}$ based on these samples?

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    do you know the population means in all the samples or do you have to estimate them as well? – 2017-02-11
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    @SlugPue: I have to estimate them as well. – 2017-02-11
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    this might be helpful: http://math.stackexchange.com/questions/1421825/estimating-the-expectation-of-a-derivative – 2017-02-11
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    @SlugPue: Thanks. Will look into that. – 2017-02-12

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