3.5.1. Mot so phan phoi ly thuyet
3.5.1.1. Phan phoi nhi thuc a) Cong thuc
Gia sir thirc hien m phep thu doc lap. Trong m6 i phep thCr bien co' A xuat hien voi xac suat P(A) = p. Khi do xac sua't d4
bien co A xuat hien dung x lan trong m phep thu duoc xac dinh bang cong thuc:
P(x) = C ; p ‘( l - p ) m-‘
Trong do: x - Cac dai luong ngSu nhien co the nhan cac gia tri 0, 1, 2, 3,...m.
(1-p) - Xac suat khong xuat hien bien co A trong m6i phep thu 1 - p = q.
C^ - Cac he sd cua phan phoi nhi thirc (to hop chap x cua tap hop gom m phan tu). No duoc xac dinh:
x!(m - x)! m !
N hu vay: P(x) = --- p ,q m”‘ x!(m - x ) !
b) Tinh chat (khong chung minh)
- Tong cac xac suat tfnh tir moi gia trj co the cua x bang 1
Chuang 3: Cac muc do cua hien tuong tliong ke 63
- Neu p = q = 0,5 co phan phoi nhi thuc ddi xung. Neu p * q co phan phoi nhi thuc khong ddi xung.
- Cac gia tri cua he sd nhj thuc CJ, luon co tinh chat ddi xung va co the bieu didn dudi dang "tain giac" Pascal.
1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1
Cac gia tri cua C„ duoc tfnh s5n va ghi trong bang.
c) Cac tham so dac trifng
- Sd binh quan: x = X = m.p - Phuong sai: a \ = m.p.q - Do lech chuan: = yjm.p.q
3.5.1.2. Phan phoi Poisson
Trong phan phoi nhi thuc khi m ldn, viec xac dinh cac gia tri P(x) rat ton kem thdi gian. De khac phuc tinh trang do, co the su dung phan phoi Poisson thay the. Dieu kien thay the mot cach hop ly la xac suat p hoac q kha be va m tuong ddi ldn.
i) Cong thuc
Dat A. = mp va luc do xac suat de’ bien co A nao do xuat hien lung x lan co the xac dinh bang cong thuc:
64 Thong ke vu ung dung trong Bun chinh. Vien thong
Trong do: e = 2,718... Co so cua lo-ga-rit
x = 0, 1, 2... Dai luong ngau nhien rdi rac
h) Cac tham so
- So binh quan: x = A. = mp - Phuong sai: a ; = A. - Do lech chuan: a* = \fk
c ) Tinh clidt
Tong cac xac suat tfnh tu moi gia tri co the cua x bang 1 I ( P ( x ) = l
x=0 (adsbygoogle = window.adsbygoogle || []).push({});
Viec xac dinh cac gia tri P (x) da duoc ghi san.
3.5.1.3. Phan phoi chuan
Phan phoi chuan su dung de xet cac dai luong ngau nhien lien tuc.
a) Phuong trinh dudng cong phan phoi chuan
Phan phoi chuan la mot phan phoi lien tuc. Tren do thi cac tung do cua dudng cong duoc xac dinh:
Chuang 3: Cac muc dp cua hien tuong tliong Ice 65
Trong do: n = 3,14159; e = 2,718;
x - Dai luong ngau nhien lien tuc co the lay moi gia tri tir - oo den + oo
x - So binh quan; g x - Do lech chuan.
Dudng cong f(x) co dang hinh chuong ddi xung. Tai gia tri x = x tung do dat gia tri cuc dai.
f(x ) = f(x ) = ---- r = e ° =
a x V271 a x y[2n
Dinh cua hinh chuong la P ung vdi tung do cuc dai. O hai ben cua P tung do giam dan ; luc dau giam ft, sau giam nhieu, den 0 2 diem ddi xung ung vdi hoanh do x - a x va x + a x la 0 2 diem uon dudng cong. Sau do tung do giam cham va cuoi ciing ben trai cung nhu ben phai dudng cong tiem can vdi true hoanh.
b) Cac tham so dac trung
- Sd binh quan: x = jx f(x )d x — 00
2
- Phuong sai: a2 = j ( x - x ) f(x)dx
—oo
- D o lech chuan: a , = y[ai
Sd binh quan xac dinh vi tri dudng cong tren true x, con do lech chuan xac djnh hinh dang nhon hay det ciia dudng cong (Do lech chuan be thi dudng cong nhon, do lech chuan ldn thi dudng cong det).
66 Tliong ke va ung dung trong Bun chinh. Vien thong
Vi dudng cong ddi xung qua x nen sd binh quan vua la sd mot. vua la sd trung vj x = M„ = M c
Ham f(x) la ham mat do xac suat nen co the’ chung minh dupe: } f(x)dx = 1
c) Dudng cong chuan thu gon
Neu trong cong thuc dat:
x - x t = ---
a .
Khi do sd binh quan theo bien mdi (t) se la: 1 = ^ = 0
a ,
Cac phuong sai theo bien t: (adsbygoogle = window.adsbygoogle || []).push({});
2 2 ^ 2 1
— C ( x ) = 1
On °*
Nhu vay tu cach bien ddi cong thurc co the viet thanh dang:
(p(t) = - J = e 2
V 2n
Ham (p(t) dupe goi la phuong trinh dudng cong c h u in thu gon, con t dupe goi la bien chu&n hoa. C) day t dupe phan phoi theo luat chudn vdi sd binh quan t = 0 va dp lech chudn a, = 1.
Chuang 3: Cac muc do cua hien tuong tliong ke 67
3.5.2. So sanh phan phoi thuc nghiem vdi phan phoi ly thuyet 3.5.2.1. So sanh bdng tieu chuan A,2
Khi so sanh phan phoi thuc nghiem voi phan phoi ly thuyet bang tieu chuan A2, ta tien hanh theo cac budc sau:
Budc 7: Ve do thi tan so thuc nghiem qua do phan doan dang phan phoi ly thuyet
Do co nhidu dang phan phoi ly thuyet, nguoi ta thudng chi chon ra mot dang de dem so sanh, vi vay de do ton thoi gian phai chon nhieu lan, ngay tir dau can co su phan doan tuong doi chinh xac dang phan phoi ly thuyet. M6i dang phan phoi ly thuyet co mot do thi bieu di6n nhat dinh. Thong qua do thi tan so' thuc nghiem se co su lien he va can cu hop ly de phan doan. Tuy nhien dieu phan doan duoc van khong the thay the duoc ket luan cuoi cung.
Budc 2: Tfnh cac tham so' can thiet cua phan phoi thuc nghiem - Xac suat p, q (doi voi phan phoi ly thuyet va phan phoi nhi thirc) - So binh quan
- Do lech chuan - Xac suat P(x) - Tan so ly thuyet fj
Cach xac dinh cac tham so nay da duoc trinh bay cu the trong phan tren.
Budc 3: Tmh gia trj tieu chuan A.2 va so sanh ket luan Gia tri tieu chuan X2 duoc tmh theo cong thuc:
68 Thong ke va Ung dung trong Buu chinh, Vien tliong
Trong do: f, - Tan so thirc nghiem f - Tan sd ly thuyet
i = 1, 2,... k - Thu tu cac to.
Clu'i y: Khi ap dung cong thuc can thoa m an di£u kien trong tat ca cac to tan sd ly thuyet cung n h u tan so thuc nghiem ddu phai ldn hon hoac bang 5. Neu chua thoa m an can phai tien hanh ghep to.
Sau khi tinh duoc X2 dem so sanh vdi gia tri A.2hing (tra duoc tir bang tfnh s£n vdi do tu do T va he so sai lam cho phep P J
- Neu X2 < >^hant, thi ket luan phan phoi thuc nghiem phu hop vdi phan phoi ly thuyet vdi he sd sai lam Ps.
- Neu X2 > A.2b.nt, thi khong co co sd ket luan phu hop cua 02 phan phoi. Luc do tarn thdi coi phan phoi thuc nghiem khong phu hop vdi phan phoi ly thuyet vdi he sd sai lam Ps.
3.5.2.2. So sanh bang tieu chuan Rom anosky
0 day cac budc tien hanh cung giong nhu khi su dung tieu chuan X2, nhung de danh gia ket qua va rut ra ket luan cuoi ciing ngudi ta khong sir dung X2hing ma dung chi tieu R.
R =
Trong do: X2 - Tinh theo cong thuc d budc 3, muc 3.5.2.1 T - Do tu do.
C huong 3: Cac muc do cua hien tuong thong ke 69
Can cur vao gia tri cua R de! ket luan:
- Neu R < 3 thi phan phoi thuc nghiem phu hop vdi phan phoi ly thuyet (adsbygoogle = window.adsbygoogle || []).push({});
- Neu R > 3 thi phan phoi thuc nghiem khong phu hop vdi phan phoi ly thuyet.
3.5.2.3. So sanh bang tieu chuan Kolmogorop
Viec so sanh cung duoc tien hanh theo ba budc, trong do budc 1 va budc 2 giong nhu khi so sanh bang tieu chuan X2.
Budc 3: Tmh gia tri tieu chuan va so sanh. Tieu chuan Kolmogorop D duoc xac dinh bang cong thuc:
D = — max n F - F Trong do: Fj - Tan so thuc nghiem cong don
Fj - Tan so ly thuyet cong don
n - Tdng cac tan sd n = X f, - X f,
Gia tri cua D tfnh theo cong thuc d budc 3, muc 3.5.2.3 duoc so sanh vdi DKing
- Neu D < DKin, thi ket luan vdi he sd sai lam Ps phan phoi thuc nghiem phu hop vdi phan phoi ly thuyet.
- Neu D > thi phan phoi thuc nghiem khong phu hop vdi phan phoi ly thuyet.
70 Thong ke va ifng dung trong Buu clunli. \ ien thong
Khi su dung tieu chuan Kolmogorop khong can phai ghep cac t6 co tan so nho hon 5.
3.5.3. Ung dung cac trudng hop
3.5.3.1. So sanh phan phoi thuc nghiem vdi pha n ph o i nh i thuc (su dung tieu chuan A2)
Vi du: So dich vu EMS co trong luong dich vu khach hang den yeu cau phuc vu tai mot buu cuc la mot dai luong ngau nhien x. Gia sir cac gia tri co the cua no la 0, 1, 2, 3, 4, 5, 6 . Quan sat 218 khach hang den yeu cau ta co so lieu sau:
Bang 3.6
So djch vu EMS So khach hang den
0 18 1 48 2 62 3 40 4 27 5 15 6 8 Cong 218
So djch vu EMS den co tuan theo phan phoi nhi thuc khong? Qua trinh so sanh duoc tien hanh theo cac budc:
Budc 1: Ve do thi tan so va phan doan dang phan phoi ly thuyet. Viec ve do thi nham muc dfch phan doan dang phan phoi ly thuyet thfch hop. Trong vf du nay phan phoi ly thuyet da duoc biet trudc nen khong can ve do thi.
C huang 3: Cac m uc do cua hien tuong thong ke 71
Budc 2: Tfnh cac tham so cua phan phoi thuc nghiem - So dich vu EMS den binh quan
I x , f , 523 x = I f , 218 = 2,399 Bang 3.7 X, f, x, f, P(x) = C m X Px.qm~* f,' = n.P(x) f i - f i ( f i - f i ) 2 ( f i - f i ) 2 fi 0 18 0 0,046656 10 8 64 6,400 1 48 48 0,186624 41 7 49 1,195 2 62 124 0,311040 68 -6 36 0,529 3 40 120 0,276480 60 -2 0 400 0,667 4 27 108 0,138240 30 -3 9 0,300 5 15 75 0,036864 8 14 196 21,778 6 8 48 0,004096 1 Cong 218 523 1,000000 218 - - k 2= 36,869
- Xac suat p la xac suat de’ xuat hien dich vu EMS trong m6 i khach hang den veu cau.
x 2,399 x = mp => p = — = ---= 0,4
m 6
q = 1 - p = 1 - 0,4 = 0,6
- Xac suat P(x) tuc la xac suat de trong m6 i lan khach han den co 0, 1, 2, 3, 4, 5, 6 dich vu EMS. No duoc tmh theo con thuc 6 muc 3.5.1.1 (a).
no
72 Thong Ice va Ung dung trong Buu chinh. \ ien tliong (adsbygoogle = window.adsbygoogle || []).push({});
P(x) = q . 0 , 4 x.0,66" Gia tri cua P(x) ducrc tmh d cot trong bang. - Tan so ly thuyet f , .
f i = n. P (x) = 218. P (x) Gia tri cua no duoc ghi d cot trong bang.
Budc 3: Tmh gia tri X2 va so sanh
De’ tfnh X2 trudc het can tfnh hieu ( f, — f , ) ; ( f, — f, ) * ; va (f, - f,): / f , . Tdng cac gia tri cot chfnh la X2. Ket qua X2 = 36,869. Khi tfnh can chu y tan so ly thuyet td cuoi cung bang 1 nho hon 5 nen phai ghep vdi td dung trudc no.
Tra bang gia tri Xbang vdi he so sai lam P„ = 0,05 va do tu do: T = k - 2 = 7 - 2 = 5
Cd: ^ = 11,07
Nhu vay X2 tfnh toan = 36,869 > A ^ d o do ket luan phan phoi thuc nghiem khong phu hop vdi phan phoi ly thuyet.
Trudng hop su dung tieu chuan so sanh khac, chang han Romanosky ta cung cd ket luan tuong tu.
3 6 , 8 6 9 - 4 , R — —•/— = --- ,— — = 11,6 > 3
V2T J 2 A
3.5.3.2. So sanh phan phoi thuc nghiem vdi phan ph oi Poisson \'i du: Khao sat phan phoi thuc nghiem thdi gian dam thoai cua 1 0 0 khach hang dudi day tuan theo phan phoi ly thuyet nao?
Chitong 3: C ac muc do cua hien tuong thong ke 73
Being 3.8
Thdi gian dam th o a i (phut)
So khach hang quan sat
Thdi gian dam th o a i (phut) So khach hang quan sat 0 2 6 10 1 10 7 6 2 14 8 3 3 20 9 2 4 18 10 1 5 13 11 1 Cong 100
Budc 1: Ve do thi tan so thirc nghiem va phan doan dang phan phoi
Qua do thi cho thay cac tan so thuc nghiem duoc phan phoi tuong doi giong dang phan phoi Poisson. VI vay tien hanh so sanh voi phan phoi nay.
Budc 2: Tmh cac tham so cua phan phoi thuc nghiem Thoi gian dam thoai binh quan:
- Zx.f, 400 , , , x = ' ' = --- = 4 phut
I f , 100
Day la tham so dac trung cua phan phoi Poisson (x = X) Being 3.9 T hdi gian dam th o a i x, So khach hang f, x, f, P(x) fi' f i - f i ' ( f i - f i ) 2 ( f i - f i ) 2 fi' 0 2 0 0,0210 2 3 9 1,000 1 10 10 0,0724 7
74 Thong ke va tfng dung trong Buu chinh. Vien thong Thdi gian dam th oa i X, So khach h an g f, Xi fi P(x) f.' f, - fi ( t - O 2 ( t - t ) 2 i 2 14 28 0,1465 15 - 1 1 0.066 3 20 60 0,1954 20 0 0 0 4 18 72 0,1954 20 - 2 4 0.200 5 13 65 0,1563 16 - 3 9 0,562 6 10 60 0,1042 10 0 0 0 7 6 42 0,0595 6 0 0 0 8 3 24 0,0298 3 3 9 0,900 9 2 18 0,0132 1 10 1 10 0,0053 11 1 11 0,0019 Cong 100 400 1,000 100 - - >.2= 2,728
- Xac suat P(x) tuc la xac suat de vdi m6i khach hang dam thoai nhan cac gia tri x = 0 , 1, 2,... 1 0, 1 1.
Gia tri P(x) duoc xac dinh theo cong thuc 4 \ e~ 4 P(x) =
x! Ket qua tfnh toan ghi trong bang: - Tan so ly thuyet f, :
f, = n.P(x) = 100.P(x) Ket qua tfnh ghi vao bang.
Chuong 3: Cac m uc dp cua hien tuong thong ke 75
Budc 3: Tinh gia tri tieu chuan va so sanh
De tinh gia tri tieu chuan X2 ta can tmh (f, — f,): (f, - f, ) 2 va (f, - f,): / f,. Ket qua ghi trong hang. O cuoi cung cho ta gia tri (adsbygoogle = window.adsbygoogle || []).push({});
X2 = 2,728.
Chu y: Theo quy cfinh tan so phai ldn hon hoac bing 5 do do ta phai ghep to 1 va 2; cac to 8,9, 10, 11 vdi do tir do T = k - 2 = 7 - 2 = 5 va Ps = 0,05 ta tim trong bang gia tri Arbang = 11,07. Nhu vay:
4 „ h = 2,728 < ^ ng = 11,07
Ket luan: Khao sat 100 khach hang cho thay thdi gian dam thoai tuan theo phan phoi Poisson.
J.5.3.3. So sanh phan phoi thuc nghiem voi phan phoi chuan Vi du: Quan sat 50 buc dien bao ta cd sd tieng trong mdi butc nhu sau:
Bang 3.10
So tie n g x, B u t dien bao f. Trj so giOra x,
1 9 -2 1 2 20 21 -2 3 5 22 2 3 - 2 5 6 24 2 5 - 2 7 10 26 2 7 - 2 9 11 28 2 9 -3 1 9 30 31 - 3 3 6 32 3 3 - 3 5 1 34 Cong 50
76 Tliong ke va Ung dung trong Bun chinh. V ien tliong
Yeu cau khao sat phan phoi thirc nghiem nay co tuan theo phan phoi chuan hay khong?
* Trudng hop so sanh hang tieu cluian X2:
Budc 1: Ve do thi va phan doan dang phan phoi ly thuyet. Trong trudng hop nay phan phoi ly thuyet (phan phoi chuSn) da duoc chon trudc. Cho nen khong can ve dd thi.
Budc 2: Tinh cac tham sd cua phan phoi thuc nghiem - Sd tieng binh quan:
_ = Z x i = L 1 5 8 = 2 7 16 I f , 50 Bang 3.11 So tie n g (phut) x, Bure
dien f, x ,f, X,2 .fi t, <p(t,) fi' f i - f i ' < f i - t ') 2
( t - f i ) 2 fi 20 2 40 800 - 2,11 0,0431 1 2 4 0,667 22 5 110 2420 - 1,52 0,1257 4 24 6 144 3456 -0 ,9 3 0,2589 8 - 2 4 0,500 26 10 260 6760 -0 ,3 4 0,3765 11 - 1 1 0,090 28 11 308 8624 0,25 0,3867 12 - 1 1 0.833 30 9 270 8100 0,84 0,2803 8 1 1 0,125 32 6 192 6144 1,43 0,1435 4 1 1 0.167 34 1 34 1156 1,85 0,0721 2 Cong 50 1358 37460 L I ' 50 - - >.*=2.382 Phuong sai: o ; = x : - x ‘ = 1—— ---27,16' = 11.53 50
C huong 3: Cac muc do cua hien tuong thong ke 77
- Do lech chuan: a x = yj\ 1,53 = 3,396