the mp test is only for a specified value of hypothesis and the UMP test is for a set of values
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The COP can be more that one since the heat transferred can be more than power required by the refrigerator but the efficiency can never be more than one since the power generated by engine cannot be more than the total heat content of the fuel.
nyquist limit is for noiseless channel and given by 2Hlog2V where V are the discrete levels and H is the Bandwidth while, shanon limit is for channel with noise and given by H log2(1+S/N) where S is the signal power and N is the noise power.
The power of a test is 1 minus the probability of a Type II error.
From rule of set difference: A \ B = {x is element of A and not element of B} This is a little of first part of the question. When we have set A, set B and finding the difference of P(A) \ P(B) or the same as P(A) - P(B). First we have to make these two power sets of A, and of B. P(A) = { {}, subset of A, other subsets of A, , , (A its self)} P(B) = { {}, subset of B, other subsets of B, , , (B its self)} These two power sets will contain what ever subsets of A, or subsets of B, but first of their elements will be {}, which will be the same. From rule of set difference, I've seen many sample shown P(A) \ P(B) = { {}, subset of A, which not subset of B, , , } The big wonder is {}, the empty set still contained in the result set P(A) \ P(B), even though {} is contained in P(B). It did not being get rid off and other elements if they contained in P(B). Many internets show the same but never explain.
Viewing the data is an easy way to see some of their characteristics such as trends, seasonality, outliers, relationship between variables (linear, quadratic, power etc).