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P (A ∪ B) P (A √ √ √ n i=1 n i=1 P( P (E ( Ei ) n i=1 Ei ) mini P (E ( Ei ) ( Ei ) maxi P (E mini P (E ( Ei ) ( Ei ) maxi P (E P (E ( Ei ) A: P (A P (A B: P (A) = 0.12 ∪ B) = 0.17 − P (AC ∩ B C ) =1 ∪ B )C (A ∪ B) = 1 − P − P (AC ∩ B C ) 1 − P (AC ∪ B C ) 1 − P (AC )P (B C ) P (AC ∩ B C ) 1 P (A ∩ B) = 0.06 ∪ B) = P (A) + P (B) − P (A ∩ B) P (B ) = P (A ∪ B ) + P (A ∩ B) − P (A) = 0.17 + 0.0.06 − 0.12 = 0.0.11 X f (x) = 1 0 ≤x ≤1 x f (t) dt = 1< x< x 0 f (x) 1 dt = [t]x0 = x x75 1 x2 −0= x F (x75 ) = 0.75 1< x< ∞ F (x75 ) = x75 x75 = 0.75 1 x3 1 x 3 ∞ −1 1 − x1 1 − x1 1 x 0 F (x) = F (x) = x 1 1 dt dt = t2 x − 1 t 1 1 = t 1 = x 1 1 − x1 = 1 − x1 p x e x (1 + e − − ) 2 − 1/ (1 + ex ) /p) − pp)) /p) p/ (1 − p p)) p)) /p (1 − p log ((1 log(p/ (1 − p)) p)) F (x) = t (1 + e t ) dt dt 2 u(t) = 1 + e u(x) − −1 du = u2 u(−∞) − e −∞ x F (x) = t 1+e 1+e−x −e −1 du = u2 ∞ du = − xp t − 1 u dt 1+e 1+e−x = ∞ F (xp ) = p 1 1+e ex ex e x = ex + 1 e x (ex + 1) − x − = − F (xp ) = p ex =p ex + 1 ex = p (ex + 1) ex = pe x + p ex pex = p ex (1 p) p) = p p ex = 1 p p xp = log 1 p p p p p p p − p p p − p − − c cxk c>1 0<x<1 k 1 k k+1 − 1 xk+1 cx dx = c k+1 k 0 0 c (1 k+1 = − 0) = k +c 1 1 1 0 x> 0 x 1 e 2 2 t/ t/2 − 0 F (x) = 1 −x/2 e x/2 2 c xk+1 k+1 = 1 ⇒ c= k +1 f (x) = c =1 k+1 1 k+1 2 1 k k+2 dt = x 2 t/ t/2 e − 0 x50 = e 0 − 2 t/ t/2 − x = e0 F (x) 2 x/ x/2 −e − F (x50 ) = 0.5 =1 −e 2 x/ x/2 − F (x50 ) = 0.5 1 −e x50 /2 = 0. 0 .5 x50 /2 = 0. 0 .5 − e − − −x50/2 = log x50 /2 = log 1 2 1 2 x50 /2 = log (2) (2) x50 = 2 log log (2 (2)) ≈ 1.386 x2 2xe x> 0 − F (x) = x t2 2te dt − F (x) 0 −t2 u(x) eu du = − F (x) = − u(t) = u(0) x50 du = x − 2 −2tdt 0 eu du = [eu ] x2 − 0 F (x50 ) = 0.5 = e0 −e x2 − =1 −e x2 − F (x50 ) = 0.5 − 1 −e 2 x50 2 x50 e − 2 x50 − = 0.5 = 0.5 = log 1 2 2 x50 = log (2) (2) x50 = log (2) ≈ 0.833 h(x) c 0 < h(x) < ∞ x = 1, 2, . . . , I c h(x) · 1 1+ I x=1 h(x) h(x) I x=1 I x=1 h(x) h(I ) 1 − 0 < c h(x) < · ∞ I · c h(x) = 1 · x c h(x) = 1 x=1 I c h(x) = 1 x=1 1 − I c= h(x) x=1
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