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d v v=h·d h h km/sM p c h [0 10) [10 20) [20 30) [30 40) [40 50) [50 60) [60 70) [70 80) [80 90) [90 100) [100 110) [0,10) [10 20) [20,30) [30,40) [40,50) [50 60) [60 70) [70 80) [80 90) [90 100) [100 110) Istogramma di H − 11 classi H [km/s * Mpc] Density 020406080100120 0.0000.0050.0100.0150.020 n= 36 pn 1 + log2n [0,20) [20 40) [40,60) [60,80) [80,10) [100 120) Istogramma di H − 6 classi H [km/s * Mpc] Density 020406080100120 0.0000.0050.0100.015 h36'(2·5+0·15 + 2·25 + 2·35 + 6·45 + 7·55 + 5·65 + 6·75 + 3·85 + 2·95 + 1·105)/36 = 58.61 Q32[70 80) 0.75'0.6667 + (Q3 70)·0,0167 =)Q3'75. X fX(x)= ( ↵(x+ 1)3, 1x1, 0, . ↵ fX FX X X X 8 >< >: ↵ 0, 1= Z1 1 ↵(x+ 1)3dx=↵(x+ 1)4 4 1 1 =)↵=1/4. 1x1 FX(x)= Zx 1 1 4(t+ 1)3dt=(x+ 1)4 16 FX(x)= 8 >< >: 0,x < 1, 1 16(x+ 1)4, 1x1, 1,x > 1. −2−1012 0 2 4 6 8 Densità x f(x) mediamediana −2−1012 0.00.20.40.60.81.0 Ripartizione x F(x) mediamediana P(X>0) = 1 P(X0) = 1 FX(0) = 1 1 16=0.9375 m>0 P(X0)<0.5 m<E[X] fX FX(m)=0.5=)1 16(m+ 1)4=1 2=)m+1=4p 8=)m=0.6817. E[X]= Z1 1 x·1 4(x+1)3dx= 1 4 x5 5+3 4 x4 4+3 4 x3 3+1 4 x2 2 1 1 =1 4 1 5+1 5 +1 4[1 + 1] =6 10=0.6. n X X⇠B(363,0.92) 363·0.08 = 29.04>5 363·0.92 = 333.96>5 X⇠B(363,0.92)'N(333.96,26.7168). P(X= 363) = 0.92363=7.2·10 14 P(X325) =P(X325.5)' ⇣325.5 333.96p 26.7168 ⌘ = ( 1.64) = 1 (1.64) = 0.0508 = 5.08% P(X325) = 0.0550 = 5.5% Y n Y⇠B(n,0.92) n P(Y325)<0.025 n>363 Y⇠B(n,0.92)'N(0.92n,0.0736n). 0.025>P(Y325) = ✓325.5 0.92np 0.0736n ◆ 325.5 0.92np 0.0736n< z0.025= 1.96 n>364.84 n= 365 P(Y>330)'1 ⇣330.5 365·0.92p 365·0.92·0.08 ⌘ =0.8467 = 84.67%. P(Y>330) = 0.8467 = 84.67%.