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Let us denote (E_{mathcal {H}_{i}}(theta _{k,l})) and (V_{mathcal {H}_{i}}(theta _{k,l})) the expectation and the variance, respectively, of the LR Λ lr u k,l ) under hypothesis (mathcal {H}_{i}, i={0,1}).
R l ≜ Ε[Vec(χ l )Vec H (χ l )] with Ε denoting the expectation and it can be estimated by the samples from the N R neighbor range bins, i.e., R ^ l = 1 N R ∑ k = 1 N R Vec χ l, k Vec H χ l, k. (36).
Expect the expectation.
The absolute value of the expectation of v i ( l 1 ) 2 v i ( l 2 ) v i ( l 3 ) and v i ( l 1 ) v i ( l 2 ) v i ( l 3 ) v i ( l 4 ) are both of order n − 3 (green and magenta).
The expectation of v i ( l 1 ) 2 v i ( l 2 ) 2 is approximately n − 2 (red).
Empirically, the expected value of v i ( l ) 4 is approximately 2 n − 5 / 3 (black); the dashed line is 2 n − 5 / 3. The absolute value of the expectation of v i ( l 1 ) v j ( l 2 ) is n − 2 if i = j (blue) and essentially 0 if i ≠ j (data not shown; the average value was 10−19 or smaller).
When L is sufficiently large, the sample average value is used to approximate the expectation value.
The expectation is plausible.
That's the expectation".
Because the expectation is zero".
The expectation level changed completely".
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com