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New PDF release: Antenna Theory: Pt. 2

By Robert E. Collin, Francis J. Zucker

ISBN-10: 0070118000

ISBN-13: 9780070118003

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7. 10. Bonferroni inequalities. Let A1 , A2 , . . An be events and A = ∪ni=1 Ai . n Show that 1A ≤ i=1 1Ai , etc. and then take expected values to conclude n P (∪ni=1 Ai ) ≤ P (Ai ) i=1 n P (∪ni=1 Ai ) ≥ P (Ai ) − i=1 n P (∪ni=1 Ai ) ≤ P (Ai ∩ Aj ) i 0 then n n p(m) ym p(m)ym ≥ m=1 m=1 When p(m) = 1/n, this says the arithmetic mean exceeds the geometric mean.

Are uncorrelated but not independent. 8. (i) Show that if X and Y are independent with distributions µ and ν then µ({−y})ν({y}) P (X + Y = 0) = y (ii) Conclude that if X has continuous distribution P (X = Y ) = 0. 9. Prove directly from the definition that if X and Y are independent and f and g are measurable functions then f (X) and g(Y ) are independent. 10. Let K ≥ 3 be a prime and let X and Y be independent random variables that are uniformly distributed on {0, 1, . . , K − 1}. For 0 ≤ n < K, let Zn = X + nY mod K.

If n |fn |dµ < ∞ then n fn dµ = n fn dµ. 2. Let g ≥ 0 be a measurable function on (X, A, µ). 3. Let F , G be Stieltjes measure functions and let µ, ν be the corresponding measures on (R, R). 7. PRODUCT MEASURES, FUBINI’S THEOREM (ii) (a,b] F (y) dG(y) + (a,b] 35 G(y) dF (y) = F (b)G(b) − F (a)G(a) + µ({x})ν({x}) x∈(a,b] (iii) If F = G is continuous then (a,b] 2F (y)dF (y) = F 2 (b) − F 2 (a). To see the second term in (ii) is needed, let F (x) = G(x) = 1[0,∞) (x) and a < 0 < b. 4. Let µ be a finite measure on R and F (x) = µ((−∞, x]).

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Antenna Theory: Pt. 2 by Robert E. Collin, Francis J. Zucker


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