Примеры решённых задач по теории вероятности, страница 2

p1 = C20+ 0 −1 ⋅ p 2 q 0 + C21+1−1 p 2 q1 +322 232 3222 22 3+C2+ 2−1 ⋅ p q + C2+ 3−1 ⋅ p q = p + 2 p q + 3 p q + 4 p q . G_h[oh^bfuc bkoh^ ohly [u \ h^ghfbkiulZgbb ± kh[ulb_ ijhlb\hiheh`gh_ hlkmlkl\bx wlh]h bkoh^Z \h \k_o bkiulZgbyo ihNwlhfm P = 1 − (1 − p1 ) .3.3.Ijhba\h^ylk_jbc[jhkZgbcfhg_lu^hi_j\h]hihy\e_gby]_j[Z\k_jbbDZdh\Z\_jhylghklvqlhjh\gh\k_jbyo]_j[\i_j\u_ihy\blkyijbk_^vfhf[jhkZgbbfhg_lu"J_r_gb_: j1 ± \_jhylghklv g_h[oh^bfh]h bkoh^Z \ h^ghc k_jbb jZkkqblu\Z_lky k ih1fhsvxnhjfmeu^ey]_hf_ljbq_kdh]hjZkij_^_e_gby p1 = pq 6 , ]^_p = q = bkdhfZy\_jh2851515ylghklv\uqbkey_lkyih[bghfbZevghcko_f_ P = C100 ⋅ p1 (1 − p1 ) .3.4. Ijhba\h^yl k_jbc [jhkZgbc b]jZevghc dhklb ^h lj_lv_]h ihy\e_gby r_klb hqdh\DZdh\Z\_jhylghklvqlhr_klvhqdh\ihy\ylky\lj_lbcjZaijbk_^vfhf[jhkZgbbdhklbohly[u\h^ghcbak_jbc"J_r_gb_: p1±\_jhylghklvmdZaZggh]hkh[ulby\h^ghck_jbb^Zggu_bkiulZgbyhib341 5ku\ZxlkyjZkij_^_e_gb_fIZkdZeykn bk = 4: p1 = C34+ 4−1 ⋅ p 3 q 4 = C64     . <_jhyl6 6ghklv ihy\e_gby gm`gh]h bkoh^Z ohly [u \ h^ghc ba k_jbc ± kh[ulb_ ijhlb\hiheh`gh_ d20hlkmlkl\bxwlh]hbkoh^Z\h\k_ok_jbyolh_klv P = 1 − (1 − p1 ) .3.5.Ijhba\h^yl k_jbx g_aZ\bkbfuo bkiulZgbc ^h ihy\e_gby i_j\h]h mki_oZ DZ`^h_bkiulZgb_ khklhbl \ n [jhkZgbyo b]jZevghc dhklb mki_ohf ijb wlhf kqblZ_lky \uiZ^_gb_ohly[uh^bgjZaljzobebq_lujzohqdh\DZdh\Z\_jhylghklvqlhmki_o\i_j\u_ijhbahc^zl\h^bggZ^pZlhfbkiulZgbbk_jbb"J_r_gb_mki_oh^gh]hbkiulZgby±kh[ulb_ijhlb\hiheh`gh_dhlkmlkl\bxmki_oZnn2jZalh_klv p1 = 1 −   ±\_jhylghklvmki_oZh^gh]hbkiulZgby>eyhij_^_e_gbybkdhfhc3\_jhylghklbijbf_gbfnhjfmem]_hf_ljbq_kdh]hjZkij_^_e_gby P = p1 (1 − p1 ) .103.6.

Ijhba\h^yl k_jbx g_aZ\bkbfuo bkiulZgbc ^h ihy\e_gby iylh]h mki_oZ DZ`^h_bkiulZgb_ khklhbl \ n [jhkZgbyo fhg_lu mki_ohf ijb wlhf kqblZ_lky \uiZ^_gb_ ohly [uh^gh]h]_j[ZDZdh\Z\_jhylghklvqlhk_jbyaZdhgqblkygZ^_\ylhfbkiulZgbb"J_r_gb_: p1 – \_jhylghklvmki_oZ\h^ghfbkiulZgbblh_klv\uiZ^_gb_]_j[Z\ohly[uh^ghcban ihiulhdwlhkh[ulb_ijhlb\hiheh`gh_d\uiZ^_gbxj_rzldb\h\k_onbkn1iulZgbyolh_klv p1 = 1 −   . <_jhylghklvihy\e_gbyiylh]hmki_oZgZ^_\ylhfbkiulZgbb2hij_^_ey_lky ih nhjfme_ ^ey jZkij_^_e_gby IZkdZey p = Cp (1 − p1 )455 + 4 −1 144n1=C   ⋅2485  1 n ⋅ 1 −    . 2 3.7.

Ijhba\h^yl k_jbx g_aZ\bkbfuo bkiulZgbc ^h ihy\e_gby i_j\h]h mki_oZ DZ`^h_bkiulZgb_khklhbl\[jhkZgbyob]jZevghcdhklb^h\lhjh]hihy\e_gbyljhcdbbebq_l\zjdbmki_ohfkqblZ_lky\lhjh_ihy\e_gb_mdZaZggh]hqbkeZhqdh\ijbiylhf[jhkZgbbdhklbDZdh\Z\_jhylghklvqlhk_jbyaZdhgqblkygZk_^vfhfbkiulZgbb"J_r_gb_\_jhylghklvp1 mki_oZh^gh]hbkiulZgbyfh`_l[ulvjZkkqblZgZihnhjfme_^ey jZkij_^_e_gby IZkdZey p1 = C33+ 2 −11  323 11 −  ; lh]^Z bkdhfZy \_jhylghklv jZkkqblu\Z 3_lkyihnhjfme_^ey]_hf_ljbq_kdh]hjZkij_^_e_gby P = p1 (1 − p1 )6632 32 =1 − .243  243 3.8. Ijhba\h^ylk_jbxg_aZ\bkbfuobkiulZgbc^hihy\e_gbyi_j\h]hmki_oZDZ`^ucwdki_jbf_glkhklhblbang_aZ\bkbfuobkiulZgbck\_jhylghklvxiheh`bl_evgh]hbkoh^Zp\dZ`^hfbkiulZgbbMki_ohfkqblZ_lkyihy\e_gb_qbkeZiheh`bl_evguobkoh^h\ij_\hkoh^ys_]hm (0 ≤ m < n). DZdh\Z\_jhylghklvqlhmki_o\i_j\u_[m^_l^hklb]gml\iylhfwdki_jbf_gl_k_jbb"J_r_gb_ \_jhylghklv mki_oZ h^gh]h bkiulZgby p1 =n∑Ck = m +1knp k q n − k .

Lh]^Z ih nhjfme_^ey]_hf_ljbq_kdh]hjZkij_^_e_gby p = p1 (1 − p1 ) .43.9. Bf__lkyRmjgR ≥ 2dZ`^Zybodhlhjuokh^_j`blni_j_gmf_jh\Zgguohl^hnrZjh\AZl_fkdZ`^hcmjghcg_aZ\bkbfhhlhklZevguoijh\h^blkyke_^mxsZyijhp_^mjZdZ`^ucrZjmjgug_aZ\bkbfhhlhklZevguorZjh\k\_jhylghklvxpm^Zey_lkybamjguZk\_jhylghklvxq = 1 – p hklZ\ey_lky\g_cImklv:s±fgh`_kl\hghf_jh\rZjh\hklZ\rboky\s-hcmjg_s = 1,…R. GZclb P {A1 ∩ ...

∩ AR = ∅}.J_r_gb_bkdhfh_kh[ulb__klvh[t_^bg_gb_n kh[ulbcdZ`^h_badhlhjuokhklhbl\lhfqlhrZjkg_dhlhjufnbdkbjh\Zggufghf_jhfm^Zey_lkybaohly[uh^ghcmjgu<_jhylghklvihke_^g_]h p1 = 1 − q R . Lh]^Zihkdhevdm\ulZkdb\Zgb_dZ`^h]hbarZjh\hkms_kl\ey_lkyg_aZ\bkbfhhlhklZevguop = np1 = n(1 – qR).Qbkeh\u_oZjZdl_jbklbdbkemqZcguo\_ebqbg4.1. GZclbfZl_fZlbq_kdb_h`b^Zgby^bki_jkbbbdh\ZjbZpbxkemqZcguo\_ebqbg ξ =2ν – µbη = ν + 3µ_kebkemqZcgu_\_ebqbguνbµg_aZ\bkbfubjZkij_^_e_gughjfZevghkiZjZf_ljZfbZ1, σ1bZ2, σ2khhl\_lkl\_gghJ_r_gb_^eyghjfZevgh]hjZkij_^_e_gbyFν = Z1, Fµ = Z2bak\hckl\fZl_fZlbq_kdh]hh`b^ZgbyFξ = 2Fν – Fµ = 2Z1 – Z2; Dξ = Mξ2 – (Mξ)2 = M(2ν – µ)2 – (2Mν – Mµ)2 = Dν +4Dµ – 4M(νµ) + 4Mν⋅Mµ = Dν + Dµ = σ 12 + 4σ 22 .

:gZeh]bqghMη = a1 + 3a2; Dη = σ 12 + 9σ 22 .D(ξ + η) = M(3ν + 2µ)2 – (M(3ν + 2µ))2 = 9Dν + 4Dµ = 9σ 12 + 4σ 22 . 2cov(ξ,η) = D(ξ + η) – Dξ –Dη = 7σ 12 − 9σ 22 ⇒ ⇒ cov (ξ ,η ) =7σ 12 − 9σ 22.24.2. GZclbfZl_fZlbq_kdb_h`b^Zgby^bki_jkbbbdh\ZjbZpbxkemqZcguo\_ebqbg ξ =3ν – 2µ b η = ν + 2µ _keb kemqZcgu_ \_ebqbgu ν b µ g_aZ\bkbfu b bf_xl [bghfbZevgu_jZkij_^_e_gbykiZjZf_ljZfbn1, p1bn2, p2khhl\_lkl\_gghJ_r_gb_^ey[bghfbZevgh]hjZkij_^_e_gbyMν = n1p1, Mµ = n2p2, Dν = n1p1(1 – p1), Dµ= n2p2(1 – p2ZgZeh]bqghMξ = 3n1p1 – 2n2p2, Dξ = 9n1p1(1 – p1) + 4n2p2(1 – p2); Mη = n1p1+ 2n2p2, Dη = n1p1(1 – p1) + 4n2p2(1 – p2); D(ξ + η) = D(4ν) = 16n1p1(1 – p1).

2cov(ξ,η) = D(ξ + η)– Dξ – Dη = 6n1p1(1 – p1) – 8n2p2(1 – p2), lh_klvcov(ξ,η) = 3n1p1(1 – p1) – 4n2p2(1 – p2).4.3. GZclbfZl_fZlbq_kdb_h`b^Zgby^bki_jkbbbdh\ZjbZpbxkemqZcguo\_ebqbg ξ =–ν – 2µbη = ν – 2µ_kebkemqZcgu_\_ebqbguνbµg_aZ\bkbfubbf_xljZkij_^_e_gbyImZkkhgZkiZjZf_ljZfbλ1bλ2khhl\_lkl\_gghJ_r_gb_^eyjZkij_^_e_gbyImZkkhgZMν = Dν = λ1, Mµ = Dµ = λ2, ZgZeh]bqghMξ= –λ1 – 2λ2, Mη = λ1 – 2λ2, Dξ = Dη = λ1 + 4λ2. D(ξ + η) = D(–4µ) = 16λ2; cov(ξ,η) = –λ1 + 4λ2.4.4. GZclbfZl_fZlbq_kdb_h`b^Zgby^bki_jkbbbdh\ZjbZpbxkemqZcguo\_ebqbg ξ =3ν – 2µ b η = ν + 2µ _keb kemqZcgu_ \_ebqbgu ν b µ g_aZ\bkbfu b bf_xl ihdZaZl_evgu_jZkij_^_e_gbykiZjZf_ljZfbλ1bλ2khhl\_lkl\_ggh+∞J_r_gb_^eyihdZaZl_evgh]hjZkij_^_e_gbyMν = 1/λ1, Mµ = 1/λ2.

Dν = λ1 ∫ x 2e − λ1x dx −0−+∞1111= 2 ∫ xe − λ1x dx − 2 = 2 ; Dµ = 2 .2λ1λ1 λ1λ20Dξ =:gZeh]bqghMξ =3 21 2− , Mη = + ,λ1 λ2λ1 λ294141634+ 2 , Dη = 2 + 2 , D (ξ + η ) = D ( 4ν ) = 2 . cov (ξ ,η ) = 2 − 2 .2λ1 λ2λ1 λ2λ1λ1 λ2EhdZevgZyij_^_evgZyl_hj_fZBgl_]jZevgZyij_^_evgZyl_hj_fZFmZ\jZEZieZkZ5.1.Kdhevdh[jhkZgbcfhg_lugm`ghijh\_klbqlh[uk\_jhylghklvxij_\hkoh^ys_chlghkbl_evgZyqZklhlZihy\e_gbc]_j[ZhlebqZeZkvhl\_jhylghklbihy\e_gby]_j[ZihZ[khexlghc\_ebqbg_g_[he__q_fgZ"J_r_gb_\_jhylghklv\uiZ^_gby]_j[Zj = 0.5; hlghkbl_evgZyqZklhlZihy\e_gbc]_j[Zµµ − npµjZ\gZ n , lh _klv lj_[m_lky hij_^_eblv P  n − 0.5 < 0.01 = P −0.01 < n< 0.01 =nn nµ − npnn n = P −0.01< n< 0.01 ihbgl_]jZevghcij_^_evghcl_hj_f_ ≈ 2Φ 0  0, 01pqpq pqnpqxFmZ\jZEZieZkZ ]^_ Φ 0 ( x ) = ∫ e−u22du. >ZggZy \_jhylghklv ^he`gZ ij_\ukblv lh _klv0bkoh^ybafhghlhgghklbN0(o), 0.01nn≥ 1.96, ihwlhfm= 4n ≥ 196 ⇒ n ≥ 49.pqpq5.2.

KemqZcgZy \_ebqbgZ µn jZkij_^_e_gZ [bghfbZevgh k iZjZf_ljZfb n, p). GZclbµijb[eb`zgghagZq_gb_z,ijbdhlhjhf\uihegy_lkykhhlghr_gb_ P  n − p > z  = 0.0026, n_kebn Zp = 0.1.µµµJ_r_gb_ihZgZeh]bbk P  n − p > z  = 1 − P  n − p ≤ z  = 1 − P − z ≤ n − p ≤ z n n nn n n≈ 1 − 2Φ 0  z≈ 3.0 ⇒ z ≈ 0.018. = 0.0026 ⇒ Φ 0  z ≈ 0.4987 ⇒ zpq pq pq5.3. KemqZcgZy \_ebqbgZ µn jZkij_^_e_gZ [bghfbZevgh k iZjZf_ljZfb n, p) GZclbµijb[eb`zgghagZq_gb_pijbdhlhjhf\uihegy_lkykhhlghr_gb_ P  n − p < z  = 0.9974, n_kebn = 900, z = 0.03.µJ_r_gb_ihZgZeh]bbkP  n − p < z = nµ − npnn n n= P − z< n<z≈ 3.0 ⇒ pq = 0.09 ⇒ p 2 − p + = 0.9974 ⇒ z ≈ 2Φ 0  zpqpq pq pqnpq1 ± 0.8+0.09 = 0 ⇒ p == 0.1; 0.9.25.4.

<mjg_rZjh\bagbo[_euoBamjguba\e_dZxl[_a\ha\jZs_gbyrZjh\DZdh\Z\_jhylghklvqlhqbkeh[_euorZjh\kj_^bgbogZoh^blkyf_`^mb"J_r_gb_ hlghkbl_evgZy qZklhlZ ihy\e_gby [_euo rZjh\ ^he`gZ gZoh^blvky f_`^mµµ b lh _klv P = P 0.45 ≤ n ≤ 0.55 = P 0.45 − p ≤ n − p ≤ 0.55 − p  = ( p = 0.5) =nnµ − npnn = P −0.05≤ n≤ 0.05 ≈ ( n = 100 ) ≈ 2Φ 0 (1) = 0.6826.pqpq npqFgh]hf_jgu_jZkij_^_e_gby6.1.

>\mf_jgZy kemqZcgZy \_ebqbgZ ξ1, ξ2 jZkij_^_e_gZ jZ\ghf_jgh \ h[eZklb D h]jZgbq_gghchkvxxbdjb\hcy = exp (–x2). GZclbiehlghklvkh\f_klgh]hjZkij_^_e_gbyp1,2(x,y)fZj]bgZevgmxiehlghklvp1(xbmkeh\gmxiehlghklv p2/1(y/x)Y\eyxlkyebkemqZcgu_\_ebqbguξ1bξ2baZ\bkbfufbbdhjj_ebjh\Zggufb"J_r_gb_hij_^_ebfiehsZ^vh[eZklbD. S ( D ) =+∞∫e− x2dx = π . Lh]^Zihko_f_jZ\gh-−∞2 12e− x, ( x, y ) ∈ D1e− xf_jgh]h jZkij_^_e_gby p1,2 ( x, y ) =  π. p1 ( x ) = ∫ p1,2 ( x, y ) dy = ∫dy =.ππ\00, ( x, y ) ∉ D − ln y 1− ln ydx = 2 ∫p1,2 ( x, y )x2ξ1 b ξ2 aZ\bkbfu gZijbf_jp2 /1 ( y / x ) == e .

p2 ( y ) =  − − ln y ππp1 ( x )0, y < 0, y > 1;11p1,2 (0,1) =≠ p1 ( 0 ) p2 (1) =⋅ 0 = 0.ππxe− x−1 − x 2 +∞2dx =e −∞ = 0 = Mξ12 ⇒ Dξ1 = Mξ12 − ( Mξ1 ) = 0, lh _klv dhwnnbpbMξ1 = ∫2 π\ π_gldhjj_eypbbρ(ξ1, ξ2g_hij_^_ezgbkm^blvhdhjj_ebjh\Zgghklbg_evay26.2. >\mf_jgZy kemqZcgZy \_ebqbgZ ξ1, ξ2 jZkij_^_e_gZ jZ\ghf_jgh \ h[eZklb D h]jZgbq_gghchdjm`ghklvxjZ^bmkZRkp_gljhf\gZqZe_dhhj^bgZlGZclbiehlghklvkh\f_klgh]hjZkij_^_e_gbyp1,2(x, y)fZj]bgZevgmxiehlghklvj1(x)bmkeh\gmxiehlghklvj2/1(y/x).Y\eyxlkyebkemqZcgu_\_ebqbguξ1bξ2aZ\bkbfufbbdhjj_ebjh\Zggufb" 1, ( x, y ) ∈ D2p1 ( x ) = ∫ p1,2 ( x, y ) dy =J_r_gb_: S ( D ) = π R , ihwlhfm p1,2 ( x, y ) =  π R 2\0, ( x, y ) ∉ D. R2 − x2 12 R2 − x2 ∫dy=, x ≤Rp ( x, y )1π R2p2 /1 ( y / x ) = 1,2= − R2 − x2 π R 2=(nmgdpby hij_^_e_p1 ( x )2 R 2 − x20, x ≥ R. 2 R2 − y 2, y ≤Rξ1 b ξ2 aZ\bkbfu gZijbf_jgZ lhevdh \gmljb D :gZeh]bqgh p2 ( y ) =  π R 20, y > R,p1,2 (0, 0 ) =14≠ p1 (0 ) ⋅ p2 (0 ) = 2 2 .2πRπ R34x ( R2 − x2 )−1 2 22x R2 − x2222 Rdx =dx =Mξ1 = ∫⋅ ( R − x ) − R = 0; Mξ1 = ∫222πππRRR3\\2 2 R1 4 Rx −R −x − R = 0 = Dξ1.

Ihwlhfm ZgZeh]bqghdhwnnbpb_gldhjj_eypbb g_ hij_^_ππ R2ezgbkm^blvhdhjj_ebjh\Zgghklbg_evay6.3. >\mf_jgZy kemqZcgZy \_ebqbgZ ξ1, ξ2 jZkij_^_e_gZ jZ\ghf_jgh \ h[eZklb D h]jZgbq_gghcdjb\ufby = ex, y = –ex ijbo ≤by = e–x, y=–e–x ijbo >GZclbiehlghklvkh\f_klgh]h jZkij_^_e_gby p1,2(x, y) fZj]bgZevgmx iehlghklv j1(x) b mkeh\gmx iehlghklvj2/1(y/x). Y\eyxlkyebkemqZcgu_\_ebqbguξ1bξ2aZ\bkbfufbbdhjj_ebjh\Zggufb"J_r_gb_ \ kbem kbff_ljbb D hlghkbl_evgh hk_c dhhj^bgZl S ( D ) =∞= 4∫ e dx = −4e−x−x ∞0= 4, 0ihwlhfm1 , ( x, y ) ∈ Dp1,2 ( x, y ) =  40, ( x, y ) ∉ D.p1 ( x ) = ∫ p1,2 ( x, y ) dy =\x ln(− y ) dx ln ( − y ) e dy e x=, −1 ≤ y < 0= , x≤0 ∫∫4242x−e− ln ( − y ) :gZeh]bqgh p2 ( y ) =  b jZ\gh ijb hk=  −xln ye−x dx ln y dy e ∫ 4 = 2 , 0 < y ≤ 1. ∫ 4 = 2 , x > 0. − e− x − ln y e− x, x≤0p ( x, y )  2ξ1 b ξ2 aZ\bkbfu gZijbf_j= xlZevguo agZq_gbyo y. p2 /1 ( y / x ) = 1,2p1 ( x )e , x > 0. 211p1,2 (0,1) = ≠ p1 ( 0 ) p2 (1) = ⋅ 0 = 0.42+∞+∞+∞000−xxxexexe 2 xxe −2 x11dx + ∫dx = − ∫ e x dx + ∫ e − x dx = 0; Mξ12 = ∫dx + ∫dx =Mξ1 = ∫24 −∞4 02−∞ 2−∞ 200+∞011e2 x dx + ∫ e−2 x dx = 0 ⇒ Dξ1 = 0, ihwlhfmdhwnnbpb_gldhjj_eypbbg_hij_^_ezgbkm∫4 −∞4 0^blvhdhjj_ebjh\Zgghklbg_\hafh`gh=−6.4.

>\mf_jgZy kemqZcgZy \_ebqbgZ ξ1, ξ2 jZkij_^_e_gZ jZ\ghf_jgh \ h[eZklb D h]jZgbq_gghcweebikhfkihemhkyfba, bbp_gljhf\gZqZe_dhhj^bgZlGZclbiehlghklvkh\f_klgh]h jZkij_^_e_gby p1,2(x, y fZj]bgZevgmx iehlghklv j1(x b mkeh\gmx iehlghklvj2/1(y/xY\eyxlkyebkemqZcgu_\_ebqbguξ1bξ2aZ\bkbfufbbdhjj_ebjh\Zggufb" 1, ( x, y ) ∈ DJ_r_gb_: S(D) = πab, ihwlhfm p1,2 ( x, y ) = π ab. p1 ( x ) = ∫ p1,2 ( x, y ) dy =\0, ( x, y ) ∉ Db 1−=∫x2a2− b 1−x2a21dx =π abx2 2 a2 − x22, x ≤aa =.

:gZeh]bqgh π a2πa0, x ≥ a2 1− 2 b2 − y 2, y ≤bp2 ( y ) =  π b 2.0, y ≥ bp2 /1 ( y / x ) = 2b a 2 − x 2 : hij_^_e_gZijb|x| ≤ a. ξ1 bξ2aZ\bkbfugZijbf_j p1,2 (0, 0 ) =1≠π ab4. Fξ1 = Dξ1 kfihwlhfmdhwnnbpb_gldhjj_eypbbg_hij_^_π abezgbkm^blvhdhjj_ebjh\Zgghklbg_evay≠ p1 ( 0 ) p2 ( 0 ) =2.