Why when we want to compute the autocorrelation function, we change the variables?

[saeddawoud]

Hello every body
I want to ask something about the covariance:
Why when we want to compute the autocorrelation function, we change the variables? e.g.: suppose n1=integral (n(t)f1(t) dt) and n2=integral (n(t)f2(t) dt), where n1 and n2 are Gaussian RV with zero mean and common variance N0/2. now the cov(n1 n2) = E(n1 n2) = E(integral(n(t)f1(t) dt integral(n(a)f2(a) da)), can't we say E(n1 n2) = E(integral(n(t)f1(t) dt integral(n(t)f2(t) dt)). and why??
hope I explain the point.

 

[bulx]

Re: Probability Question

It is becuase we want to compare the values of both functions at all possible time combinations, not only when they are equal. If we use only one variable, we can see the value of both functions only at the same time.

lets consider discrete time, since it is easier to talk then.

When we want to evaluate covariance, we want to find the similarity (covariance is such a measure) between the value of f1 at time 0, and values of f2 from time 0 to ∞, then value of f1 at time 1, and values of f2 from time 0 to ∞, and so on.. Now, if we put t as the variable for both f1 and f2, the covariance calculation integral will be evaluated only for both f1 anf f2 at t=0, t =1 , t=2 etc etc.

Thus in general the covariance will be a graphically a 3D surface, with t1 on X axis and t2 on y axis.

-b

 

[saeddawoud]

Re: Probability Question

Thank you

 

[xakker]

Re: Probability Question

very gud explanation