By Bert B. Williams, John L. Gidley, Robert S. Schechter
SPE Monograph quantity 6. This monograph has been ready to function a uncomplicated reference for either the sector and learn engineer drawn to constructing a basic figuring out of the acidizing procedure and systems for designing acid remedies to extend good productiveness.
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Additional resources for Acidizing Fundamentals
If we had a sample of N pairs (xi, yi) from the bivariate population we could calculate an average value of g from g = ~ g(xi, Yi)/N Now, we will suppose that the data have been grouped into cells. Suppose that the x variate has been divided into 'a' categories and the y variate into 'b' categories. Then b g ~-- ~ E g(xi, yj) fx,,y/N i=1 j=1 where (xi,yj) are the coordinates of the mid-point of the cell and fx,,y~is the number of data in that cell, that is, the frequency. The approximation arises because we assume that all the pairs in the cell are at the mid-point.
XN } is a random sample from a distribution with mean/Zx and variance tr2x then EXi has mean Nttx, variance No2x, and standard deviation (V'N)trx. Division by n shows that ,(" has a mean ~ x and standard deviation trx/V'N. We should remember that random sampling justifies assuming independence. We notice that these results do not tell us the form of the distributions. However, the normal distribution often comes to our aid. First, under very general conditions, a linear combination of normal random variables is itself normal.
9. 18) The mean and variance of Y are defined in the same way. 3 Fig. This is approximately f(xi, y,)&xiAy, If X and Y are independent then Cov(X, Y) equals zero. The proof follows immediately from the definitions, namely cO oo Cov(X, Y) = J J (x- IZx)(y- tXr)fxr(x,y) dx dy --0o ~ oo and assuming independence O0 OO = J (x- tZx)fx(X) dx J (y- tzv)fv(Y) dy = --oo (0)(0) = 0 --o0 However, if Cov(X, Y) equals zero, X and Y are not necessarily independent. As an example, we can consider a uniform distribution of probability over a unit circle.