Teil a (2) (%i1) "Parameter der Binomialverteilung"$ (%i2) bin_n : 5$ (%i3) bin_p : 0.166666667$ (%i4) bin_q : 1-bin_p; (%o4) 0.833333333 (%i5) "Erwartungswert ew"$ (%i6) bin_ew : bin_n * bin_p; (%o6) 0.833333335 (%i7) "Standardabweichung sigma"$ (%i8) bin_sigma : ev(sqrt(bin_n * bin_p * bin_q),numer); (%o8) 0.833333334 (%i9) sumBin(n,links,rechts,p) := block( sum_ : 0, q_ : 1 - p, if (links < 0) then (links : 0), if (rechts > n) then (rechts : n), for j_: links thru rechts do ( sum_ : sum_ + p^j_ * q_^(n-j_) * binomial(n,j_) ), print("P(",links, "<= X <=", rechts,") = ",sum_) ); (%o9) sumBin(n, links, rechts, p) := block(sum_ : 0, q_ : 1 - p, if links < 0 then links : 0, if rechts > n then rechts : n, j_ n - j_ for j_ from links thru rechts do sum_ : sum_ + p q_ binomial(n, j_), print("P(", links, "<= X <=", rechts, ") = ", sum_)) (%i10) "P(ew-sigma <= X <= ew+sigma) (1-Sigma-Umgebung)"$ (%i11) sumBin(bin_n,ceiling(bin_ew-bin_sigma), floor(bin_ew+bin_sigma), bin_p)$ P( 1 <= X <= 1 ) = 0.40187757217721 (%i12) "P(ew-2*sigma <= X <= ew+2*sigma) (2-Sigma-Umgebung)"$ (%i13) sumBin(bin_n,ceiling(bin_ew-2*bin_sigma), floor(bin_ew+2*bin_sigma), bin_p)$ P( 0 <= X <= 2 ) = 0.96450617264661 (%i14) "P(ew-3*sigma <= X <= ew+3*sigma) (3-Sigma-Umgebung)"$ (%i15) sumBin(bin_n,ceiling(bin_ew-3*bin_sigma), floor(bin_ew+3*bin_sigma), bin_p)$ P( 0 <= X <= 3 ) = 0.9966563785751 (%i16) xy:makelist([x,binomial(bin_n, x)*bin_p^x*bin_q^(bin_n-x)],x,0,bin_n); (%o16) [[0, 0.40187757121271], [1, 0.40187757217721], [2, 0.16075102925669], [3, 0.032150205928498], [4, 0.0032150206005658], [5, 1.2860082433127569E-4]] (%i17) plot2d([discrete,xy],[gnuplot_curve_titles,"title 'Binomialverteilung (n=5, p=0.166666667)'"],[gnuplot_curve_styles,["with boxes fill solid 0.3"]])$
Teil a (3) (%i18) "P(k1 <= X <= k2) = P(1 <= X <= 5 )"$ (%i19) sumBin(5,1,5,0.166666667)$ P( 1 <= X <= 5 ) = 0.59812242878729 (%i20) "P(k1 <= X <= k2) = P(0 <= X <= 0 )"$ (%i21) sumBin(5,0,0,0.166666667)$ P( 0 <= X <= 0 ) = 0.40187757121271 (%i22) 0.40187757121271 + 0.59812242878729; (%o22) 1.0 Teil b (1) - (3) (%i23) "P(k1 <= X <= k2) = P(0 <= X <= 21 )"$ (%i24) sumBin(100,0,21,0.166666667)$ P( 0 <= X <= 21 ) = 0.89981652772556 (%i25) "P(X = k) = P(X = 25 )"$ (%i26) sumBin(100,25,25,0.166666667)$ P( 25 <= X <= 25 ) = 0.0098258820675571 (%i27) "P(k1 <= X <= k2) = P(21 <= X <= 100 )"$ (%i28) sumBin(100,21,100,0.166666667)$ P( 21 <= X <= 100 ) = 0.15188785007575 (%i29) "P(k1 <= X <= k2) = P(0 <= X <= 20 )"$ (%i30) sumBin(100,0,20,0.166666667)$ P( 0 <= X <= 20 ) = 0.84811214992425 (%i31) 0.15188785007575 + 0.84811214992425; (%o31) 1.0 Teil d (%i32) "Parameter der Binomialverteilung"$ (%i33) bin_n : 300$ (%i34) bin_p : 0.1666666667$ (%i35) bin_q : 1-bin_p; (%o35) 0.8333333333 (%i36) "Erwartungswert ew"$ (%i37) bin_ew : bin_n * bin_p; (%o37) 50.00000001 (%i38) "Standardabweichung sigma"$ (%i39) bin_sigma : ev(sqrt(bin_n * bin_p * bin_q),numer); (%o39) 6.454972244195426 (%i40) "P(ew-sigma <= X <= ew+sigma) (1-Sigma-Umgebung)"$ (%i41) sumBin(bin_n,ceiling(bin_ew-bin_sigma), floor(bin_ew+bin_sigma), bin_p)$ P( 44 <= X <= 56 ) = 0.68627306915098 (%i42) "P(ew-2*sigma <= X <= ew+2*sigma) (2-Sigma-Umgebung)"$ (%i43) sumBin(bin_n,ceiling(bin_ew-2*bin_sigma), floor(bin_ew+2*bin_sigma), bin_p)$ P( 38 <= X <= 62 ) = 0.94769997967356 (%i44) "P(ew-3*sigma <= X <= ew+3*sigma) (3-Sigma-Umgebung)"$ (%i45) sumBin(bin_n,ceiling(bin_ew-3*bin_sigma), floor(bin_ew+3*bin_sigma), bin_p)$ P( 31 <= X <= 69 ) = 0.99745127445438 (%i46) "Aufgabenstellung: k = 70 mal die 6 => außerhalb der 3-Sigma-Umgebung!"$
Teil e (1)/(3) (%i47) "rechtsseitiger Hypothesentest (Binomial verteilte Zufallsvaribale X)"$ (%i48) "Stichprobenumfang n = 100"$ (%i49) "Irrtumswahrscheinlichkeit alpha = 0.05"$ (%i50) "Nullhypthese H0: p <= 0.1"$ (%i51) "Prüfvariable X ist B(100;0.10)-verteilt"$ (%i52) "Konstruktion des Ablehnungsbereiches K = {gr;...;100}"$ (%i53) "K = {16;...;100}"$ (%i54) "=> Annahmebereich Kquer = {0;...;15}"$ (%i55) "Der Würfel zeigt 20-mal die Sechs => Nullhypothese H0 verwerfen!"$