IqvjUb †Kv¤úvwb wj. wewfbœ ai‡bi wUwf, wd«R I Ab¨vb¨ B‡jKUªwb· mvgMÖx Drcv`bKvix cÖwZôvb| GwU ¯^ímg‡qi g‡a¨ †`‡k mycwiwPwZ jvf K‡i‡Q| GwU †µZv‡`i iæwP I cQ›` Abyhvqx wewfbœ g‡W‡ji cY¨ mvgMÖx Drcv`b K‡i| ZvQvov cÖwZôvbwU m‡e©vËg weµ‡qvËi †mev wbwðZ K‡i| Gi gva¨‡g GwU h‡_ó mybvg AR©‡b mÿg n‡q‡Q| mKj †kÖwYi †µZv‡`i Kv‡Q cÖwZôvbwUi Drcvw`Z mvgMÖx e¨vcK RbwcÖqZv cvIqvq Zv we‡ePbvq †i‡L c‡Y¨i g~j¨ wba©viY K‡i‡Q| cÖwZôvbwU C`-Dj-Avhnv Dcj‡ÿ 10% Qvo †NvlYv Ki‡Q| hv‡Z K‡i GwU Gi cÖwZ‡hvMx‡`i mv‡_ †gvevwejv K‡i †ewk wewµ Ki‡Z mÿg n‡Z cv‡i|
K. †mev cY¨ Kx?
L. DËg µq weµ‡qi A‡a©K Ñ e¨vL¨v K‡iv|
M. DÏxc‡K IqvjUb †Kv¤úvwb wj. †Kvb c×wZ‡Z c‡Y¨i g~j¨ wba©viY K‡i| e¨vL¨v K‡iv|
N. DÏxc‡K ÔIqvjUbÕ †Kv¤úvwb wj. KZ…©K C`-Dj-Avhnv Dcj‡ÿ c‡Y¨i g~j¨ Qvo †`Iqv cÖwZ‡hvwMZv †gvKvwejvq KZUzKz Kvh©Ki f~wgKv ivL‡e e‡j Zzwg g‡b K‡iv? Dˇii mc‡ÿ hyw³ `vI|
M †Kv¤úvwbwU f¨vjy wfwËK g~j¨ wba©viY K‡i|
N weµq cÖmv‡ii msÁv w`‡q cÖwZ‡hvwMZv ‡gvKv‡ejvq Gi f~wgKv Av‡jvPbv Ki‡Z n‡e|
--- welq : cwimsL¨vb wØZxq cÎ
m„Rbkxj cÖkœ
2q Aa¨vq (‰`e PjK I m¤¢vebv web¨vm)
## ˆ`e PjK : †h Pj‡Ki cÖwZwU gv‡bi GKwU wbw`©ó m¤¢vebv _v‡K, Zv‡K ˆ`e PjK e‡j|
A_©vr m¤¢vebvhy³ PjK‡K ˆ`e PjK e‡j|
g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … P(xn) †hLv‡b, P(xi) = 1.
ˆ`e PjK `yÕwU kZ© †g‡b P‡j : (i) P(x) ≥ 0 (ii) P(x) = 1.
ˆ`e PjK `yB cÖKvi| h_v : (i) wew”Qbœ ˆ`e PjK (ii) Awew”Qbœ ˆ`e PjK
## m¤¢vebv A‡cÿK : †Kv‡bv wew”Qbœ ˆ`e Pj‡Ki m¤¢vebv †h MvwYwZK m~‡Îi mvnv‡h¨ wbY©q Kiv nq Zv‡K m¤¢vebv A‡cÿK e‡j|
Bnv‡K P(x) Øviv cÖKvk Kiv nq| m¤¢vebv A‡cÿK `yÕwU kZ© †g‡b P‡j : (i) P(x) ≥ 0 (ii) P(x) = 1.
## m¤¢vebv web¨vm : †Kv‡bv wew”Qbœ ˆ`e Pj‡Ki cÖwZwU gvb Ges Zv‡`i m¤¢vebv‡K †h mviYx†Z Dc¯’vcb Kiv nq Zv‡K m¤¢vebv web¨vm e‡j|
g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … P(xn) †hLv‡b, P(xi) = 1.
m¤¢vebv web¨vm `yÕwU kZ© †g‡b P‡j : (i) P(x) ≥ 0 (ii) P(x) = 1.
ˆ`e PjK x Gi m¤¢vebv web¨vm wb¤œiƒc :
## m¤¢vebv NbZ¡ A‡cÿK : †Kv‡bv Awew”Qbœ ˆ`e Pj‡Ki gvb GKwU wbw`©ó mxgvi g‡a¨ _vKvi m¤¢vebv †h MvwYwZK m~‡Îi mvnv‡h¨
wbY©q Kiv nq Zv‡K m¤¢vebv NbZ¡ A‡cÿK e‡j| Bnv‡K f(x) Øviv cÖKvk Kiv nq|
x x1 x2 x3 … … xn
P(x) P(x1) P(x2) P(x3) … … P(xn)
m¤¢vebv NbZ¡ A‡cÿK `yÕwU kZ© †g‡b P‡j : (i) f(x) ≥ 0 (ii) ∫ f(x)dxab = 1.
## web¨vm A‡cÿK : †Kv‡bv ˆ`e Pj‡Ki me©wb¤œ gvb n‡Z †Kv‡bv wbw`©ó gvb ch©šÍ mKj m¤¢vebvi mgwó wbiæcYKvix dvskb‡K web¨vm dvskb ev web¨vm A‡cÿK e‡j| Bnv‡K F(x) Øviv cÖKvk Kiv nq|
web¨vm A‡cÿ‡Ki ag© ev ˆewkó¨ :
(i) a < b n‡j P(a ≤ x ≤ b) = F(b) – F(a)
(ii) ‰`e Pj‡Ki `ywU gvb a ≤ b Ges 0 ≤ F(x) ≤ 1 nq, Z‡e F(a) ≤ F(b) n‡e|
(iii) F(x) cwimxgvi ev‡g k~b¨ Ges Wv‡b GK nq| A_©vr F(-∞) = 0 I F(∞) = 1.
(iv) x GKwU wewPQbœ ˆ`e PjK Ges xi-1 < xi n‡j P(xi) = F(xi) – F(xi-1) (v) x GKwU wewPQbœ ˆ`e PjK Ges me©‡kl gvb xn n‡j P(xn) = 1 .
## hy³ m¤¢vebv A‡cÿK : `ywU wew”Qbœ ˆ`e Pj‡Ki m¤¢vebv †h MvwYwZK m~‡Îi mvnv‡h¨ wbY©q Kiv nq Zv‡K hy³ m¤¢vebv A‡cÿK e‡j| Bnv‡K P(xi , yj ) Øviv cÖKvk Kiv nq| hy³ m¤¢vebv A‡cÿK `yÕwU kZ© †g‡b P‡j : (i) P(xi , yj) ≥ 0 (ii) P(xi, yj) = 1.
mgm¨v (1) : ˆ`e PjK x Gi m¤¢vebv web¨vm wb¤œiyc :
(i) k Gi gvb wbY©q Ki|
(ii) hw` y = 3x + 2 nq Z‡e y Gi m¤¢vebv web¨vm wbY©q Ki|
(iii) P(-1≤ x ≤ 2) , P( -1< x ≤ 2) , P( -1 < x < 2) , P( 0 ≤ x < 4) wbY©q Ki|
(iv) web¨vm A‡cÿK wbY©q Ki Ges P( 0< x ≤ 2) I F(2) – F(0) Gi g‡a¨ Zyjbv Ki|
mgvavb : †`qv Av‡Q, ˆ`e PjK x Gi m¤¢vebv web¨vm wb¤œiƒc :
(i) k Gi gvb wbY©q:
Avgiv Rvwb, m¤¢vebv web¨v‡mi ag©vbymv‡i, P(x) = 1
0.2 + 0.12 + 0.1 + k + 0.25 + 0.15 = 1 k + 0.82 = 1
k = 1- 0.82 k = 0.18
GLb, k Gi gvb ewm‡q ˆ`e PjK x Gi m¤¢vebv web¨vm n‡e wb¤œiyc :
(ii) †`qv Av‡Q, y = 3x + 2
x = -3 n‡j , y = 3(-3) + 2 = -9 + 2 = -7 Ges P(y) = 0.2 x = -1 n‡j , y = 3(-1) + 2 = -3 + 2 = -1 Ges P(y) = 0.12 x = 0 n‡j , y = 3(0) + 2 = 0 + 2 = 2 Ges P(y) = 0.1 x = 1 n‡j , y = 3(1) + 2 = 3 + 2 = 5 Ges P(y) = 0.18 x = 2 n‡j , y = 3(2) + 2 = 6 + 2 = 8 Ges P(y) = 0.25
x = 4 n‡j , y = 3(4) + 2 = 12 + 2 = 14 Ges P(y) = 0.15 ˆ`e PjK y Gi m¤¢vebv web¨vm wb¤œiyc :
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 K 0.25 0.15
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 K 0.25 0.15
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 0.18 0.25 0.15
(iii) ■ P(-1≤ x ≤ 2) = P(x= -1) + P(x=0) + P(x=1) + P(x= 2) = 0.12 + 0.1 + 0.18 + 0.25 = 0.65 ■ P(-1< x ≤ 2) = P(x=0) + P(x=1) + P(x= 2)
= 0.1 + 0.18 + 0.25 = 0.53 ■ P(-1< x < 2) = P(x=0) + P(x=1)
= 0.1 + 0.18 = 0.28
■ P( 0 ≤ x < 4) = P(x=0) + P(x=1) + P(x= 2) = 0.1 + 0.18 + 0.25 = 0.53 (iv) ˆ`e PjK x Gi web¨vm A‡cÿK wbY©q :
GLv‡b P( 0< x ≤ 2) = P(x = 1) + P(x = 2) = 0.18 + 0.25 = 0.43
Ges F(2) – F(0) = 0.85- 0.42 = 0.43
P( 0< x ≤ 2) = F(2) – F(0)
3q Aa¨vq (MvwYwZK cÖZ¨vkv ) ## ˆ`e PjK : m¤¢vebvhy³ PjK‡K ˆ`e PjK e‡j|
## MvwYwZK cÖZ¨vkv : †Kv‡bv ˆ`e PjK Ges Zvi wbR wbR m¤¢vebvi ¸Yd‡ji mgwó‡K H ˆ`e Pj‡Ki MvwYwZK cÖZ¨vkv e‡j|
g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … P(xn) †hLv‡b, P(xi) = 1.
ˆ`e PjK x Gi MvwYwZK cÖZ¨vkv E(x) n‡j,
E(x) = x1P(x1) + x2P(x2) + x3P(x3) + … … … + xnP(xn) = xiP(xi)
## MvwYwZK cÖZ¨vkvi ag©mg~n :
(i) †Kv‡bv aªæe‡Ki MvwYwZK cÖZ¨vkv aªæeKB nq| A_©vr , a GKwU aªæeK n‡j, E(a) = a (ii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, E(ax) = a E(x).
(iii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, E(x + a) = E(x) + a (iv) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, E(x a) = E(x) a
(v) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, E(ax + b) = a E(x) + b (vi) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, E(ax b) = a E(x) b (vii) x I y `yÕwU ‰`e PjK n‡j, E(x + y) = E(x) + E(y)
(viii) x I y `yÕwU ¯^vaxb ‰`e PjK n‡j, E(x y) = E(x) E(y)
†QvU cÖgvYmg~n :
(i) †Kv‡bv aªæe‡Ki MvwYwZK cÖZ¨vkv aªæeKB nq| A_©vr , a GKwU aªæeK n‡j, E(a) = a cÖgvY : g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i
y -7 -1 2 5 8 14
P(y) 0.2 0.12 0.1 0.18 0.25 0.15
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 0.18 0.25 0.15
F(x) 0.2 0.32 0.42 0.60 0.85 1
m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … , P(xn) †hLv‡b, P(xi) = 1 Ges a GKwU aªæeK|
Avgiv Rvwb, ˆ`e PjK x Gi MvwYwZK cÖZ¨vkv , E(x) = xi P(xi)
GLb , E(a) = a P(xi) = a P(xi) = a 1 = a [ P(xi) = 1 ] E(a) = a (Proved)
(ii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, E(a x) = a E(x).
cÖgvY: g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … , P(xn) †hLv‡b, P(xi) = 1 Ges a GKwU aªæeK|
Avgiv Rvwb, ˆ`e PjK x Gi MvwYwZK cÖZ¨vkv , E(x) = xi P(xi) GLb , E(a x) = a xi P(xi)
= a xi P(xi) = a E(x) E(a x) = a E(x) (Proved)
(iii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, E(x + a ) = E(x) + a
cÖgvY: g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … , P(xn) †hLv‡b, P(xi) = 1Ges a GKwU aªæeK|
Avgiv Rvwb, ˆ`e PjK x Gi MvwYwZK cÖZ¨vkv , E(x) = xi P(xi) GLb , E(x + a) = (xi + a ) P(xi)
= xi P(xi) + a P(xi)
= xi P(xi) + a P(xi) = E(x) + a 1 = E(x) + a E (x + a) = E(x) + a (Proved)
(iv) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, E(a x + b) = a E(x) + b
cÖgvY: g‡b Kwi, †Kv‡bv ˆ`e PjK x Gi n msL¨K gvb x1, x2 , x3 , … … … xn hv‡`i m¤¢vebv h_vµ‡g P(x1) , P(x2) , P(x3) , … … … , P(xn) †hLv‡b, P(xi) = 1 Ges a I b `yÕwU aªæeK|
Avgiv Rvwb, ˆ`e PjK x Gi MvwYwZK cÖZ¨vkv , E(x) = xi P(xi) GLb , E(a x + b) = (a xi + b ) P(xi)
= a xi P(xi) + b P(xi) = a xi P(xi) + b P(xi)
= a E(x) + b 1 = a E(x) + b E (ax + b) = a E(x) + b (Proved)
## MvwYwZK Mo I MvwYwZK cÖZ¨vkvi g‡a¨ cv_©K¨ :
MvwYwZK Mo MvwYwZK cªZ¨vkv
1) Pj‡Ki gvb ¸‡jvi mgwó‡K ‡gvU Z_¨ msL¨v Øviv fvM Ki‡j †h gvb cvIqv hvq Zv‡K MvwYwZK Mo e‡j|
†Kv‡bv ˆ`e Pj‡Ki cÖwZwU gvb I Zvi wbR wbR m¤¢vebvi ¸Yd‡ji mgwó‡K H ˆ`e Pj‡Ki MvwYwZK cÖZ¨vkv e‡j|
2) Bnv‡K x ̅ Øviv cÖKvk Kiv nq| Bnv‡K E(x) Øviv cÖKvk Kiv nq|
3) MvwYwZK Mo , x ̅ = ∑ fixi
N MvwYwZK cÖZ¨vkv , E(x) = xi P(xi) 4) Pj‡Ki cÖwZwU gv‡bi †Kv‡bv m¤¢vebv _v‡K bv| ˆ`e Pj‡Ki cÖwZwU gv‡bi m¤¢vebv _v‡K|
5) MYmsL¨v web¨vm n‡Z MvwYwZK Mo wbY©q Kiv nq| m¤¢vebv web¨vm n‡Z MvwYvZK cÖZ¨vkv wbY©q Kiv nq|
## ˆ`e Pj‡Ki †f`vsK : †Kv‡bv ˆ`e PjK Ges Zvi cÖZ¨vwkZ gv‡bi we‡qvMd‡ji e‡M©i cÖZ¨vwkZ gvb‡K H ˆ`e Pj‡Ki †f`vsK e‡j|
g‡b Kwi , x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) myZivs x ‰`e Pj‡Ki †f`vsK , V(x) = E{ x − E(x)}2 = E(x2) {E(x)}2
## †f`vs‡Ki ag©mg~n :
(i) †Kv‡bv aªæe‡Ki ‡f`vsK k~b¨ nq| A_©vr , a GKwU aªæeK n‡j, V(a) = 0 (ii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, V(ax) = a2 V(x).
(iii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, V(x + a) = V(x) (iv) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, V(x a) = V(x)
(v) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, V(ax + b) = a2 V(x) (vi) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, V(ax b) = a2 V(x) (vii) x I y `yÕwU ¯^vaxb ‰`e PjK n‡j, V(x + y) = V(x) + V(y)
(viii) x I y `yÕwU ¯^vaxb ‰`e PjK n‡j, V(x y) = V(x) + V(y)
†QvU cÖgvYmg~n :
(i) †Kv‡bv aªæe‡Ki ‡f`vsK k~b¨ nq| A_©vr , a GKwU aªæeK n‡j, V(a) = 0
cÖgvY : g‡b Kwi, x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) Ges a GKwU aªæeK|
†f`vs‡Ki msÁvbymv‡i , V(x) = E{ x − E(x)}2 Ges MvwYwZK cÖZ¨vkvi ag©vbymv‡i , E(a) = a
GLb , V(a) = E{ a − E(a)}2
= E { a a }2 [ E(a) = a ] = E { 0 }2 = E { 0 } = 0
V(a) = 0 (Proved)
(ii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, V(ax) = a2 V(x).
cÖgvY : g‡b Kwi, x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) Ges a GKwU aªæeK|
†f`vs‡Ki msÁvbymv‡i, V(x) = E{ x − E(x)}2 Ges MvwYwZK cÖZ¨vkvi ag©vbymv‡i, E(ax) = a E(x)
GLb , V(ax) = E{ ax − E(ax)}2
= E{ ax a E(x)}2 [ E(ax) = a E(x) ] = E[ a { x − E(x) }]2 = E[a2 { x − E(x) }2 ] = a2 E{ x − E(x)}2 = a2 V(x)
V(ax) = a2 V(x) (Proved)
(iii) x GKwU ‰`e PjK Ges a GKwU aªæeK n‡j, V(x + a ) = V(x)
cÖgvY : g‡b Kwi, x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) Ges a GKwU aªæeK|
†f`vs‡Ki msÁvbymv‡i , V(x) = E{ x − E(x)}2 Ges MvwYwZK cÖZ¨vkvi ag©vbymv‡i , E(x + a) = E(x) + a
GLb , V(x + a) = E{(x + a) − E(x + a)}2
= E {x + a E(x) − a }2 [E(x + a) = E(x) +a ] = E {x E(x)}2 = V (x)
V(x + a ) = V(x) (Proved)
(iv) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, V(ax + b) = a2 V(x)
cÖgvY : g‡b Kwi, x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) Ges a I b `yÕwU aªæeK|
†f`vs‡Ki msÁvbymv‡i , V(x) = E{ x − E(x)}2
Ges MvwYwZK cÖZ¨vkvi ag©vbymv‡i , E(ax + b) = a E(x) + b GLb , V(ax + b) = E{( ax + b ) − E(ax + b)}2
= E{ ax + b a E(x) − b}2 [E(ax +b) = a E(x) + b]
= E[ a { x − E(x) }]2 = E[a2 { x − E(x) }2 ] = a2 E{ x − E(x)}2 = a2 V(x)
V(ax + b) = a2V(x) (Proved)
(v) x GKwU ‰`e PjK Ges a I b `yÕwU aªæeK n‡j, V(ax b) = a2 V(x)
cÖgvY : g‡b Kwi, x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) Ges a I b `yÕwU aªæeK|
†f`vs‡Ki msÁvbymv‡i, V(x) = E{ x − E(x)}2
Ges MvwYwZK cÖZ¨vkvi ag©vbymv‡i, E(ax b) = a E(x) b GLb, V(ax + b) = E{( ax + b ) − E(ax + b)}2
= E{ ax b a E(x) + b}2 [E(ax b) = a E(x) b]
= E[ a { x − E(x) }]2 = E[a2 { x − E(x) }2 ] = a2 E{ x − E(x)}2 = a2 V(x)
V(ax b) = a2V(x) (Proved)
(v) x GKwU ‰`e PjK n‡j, E( x2 ) ≥ { E(x) }2
cÖgvY : g‡b Kwi, x GKwU ˆ`e PjK hvi MvwYwZK cÖZ¨vkv E(x) Ges a I b `yÕwU aªæeK|
†f`vs‡Ki msÁvbymv‡i , V(x) = E{ x − E(x)}2 †h‡nZz †f`vs‡Ki gvb FYvZ¥K n‡Z cv‡i bv , myZivs V(x) ≥ 0
E{ x − E(x)}2 ≥ 0
E [ x2 2 x E(x) + {E(x)}2] ≥ 0 E ( x2 ) 2 E(x) E(x) + {E(x)}2 ≥ 0 E ( x2 ) 2{E(x)}2 + {E(x)}2 ≥ 0 E ( x2 ) {E(x)}2 ≥ 0
E( x2 ) ≥ { E(x) }2 (Proved )
MvwYwZK mgm¨vejx :
mgm¨v (1) : ivBmv GKwU wbi‡cÿ gy`ªv 3 evi wb‡ÿc Ki‡jv | Dc‡ii wc‡V cÖvß †n‡Wi msL¨vi PjK x n‡j ,
(i) ‰`e PjK x Gi Mo I †f`vsK wbY©q Ki
(ii) ivBmv ej‡jv , E(x3 - 5 x2 + 3x - 10 ) Gi gvb FYvZ¥K - gšÍe¨ Ki|
mgvavb : GKwU wbi‡cÿ gy`ªv 3 evi wb‡ÿ‡ci bgybv‡ÿÎ :
S = { HHH, HHT, HTH, THH, HTT, THT, TTH, TTT } bgybv‡ÿ‡Îi †gvU Dcv`vb msL¨v, n(S) = 8
Dc‡ii wc‡V cÖvß †n‡Wi msL¨vi PjK x n‡j , x Gi gvbmg~n n‡e 0, 1, 2, 3.
GLb, ˆ`e PjK x Gi m¤¢vebv web¨vm n‡e wb¤œiyc :
x 0 1 2 3
P(x) 1
8
3 8
3 8
1 8
(i) ˆ`e PjK x Gi Mo , E(x) = x P(x) = 0 1
8 + 1 3
8 + 2 3
8 + 3 1
8 = 0 + 3
8+ 6
8+ 3
8 = 12
8 = 3
2 = 1.5
(ii) ˆ`e PjK x Gi ‡f`vsK , V(x) = E(x2) - { E(x)}2 GLv‡b , E(x2) = x2 P(x)
= (0)2 1
8 + (1)2 3
8 + (2)2 3
8 + (3)2 1
8 = 0 + 3
8 + 12
8 + 9
8 = 3 GLb, †f`vsK V(x) = E(x2) - { E(x)}2
= 3 - (1.5)2 = 3 - 2.25 = 0.75 (iii) GLv‡b , E(x) = 1.5 Ges E(x2) = 3
E(x3) = x3 P(x) = (0)3 1
8 + (1)3 3
8 + (2)3 3
8 + (3)3 1
8 = 0 + 3
8 + 24
8 + 27
8 = 54
8 = 6.75 GLb, E(x3 - 5 x2 + 3x - 10 ) = E(x3) - 5 E(x2) + 3 E(x) - 10 = 6.75 - 5 3 + 3 1.5 - 10 = 6.75 - 15 + 4.5 - 10 = - 13.75 , hv FYvZ¥K msL¨v|
myZivs ivBmvi gšÍe¨ mwVK|
mgm¨v (2) : ˆ`e PjK x Gi m¤¢vebv web¨vm wb¤œiyc :
(i) k Gi gvb wbY©q Ki|
(ii) ˆ`e PjK x Gi Mo I †f`vsK wbY©q Ki|
(iii) hw` y = 3x + 2 nq Z‡e y Gi m¤¢vebv web¨vm wbY©q Ki|
(iv) E(y) I V(y) Gi gvb wbY©q Ki|
mgvavb : †`qv Av‡Q, ˆ`e PjK x Gi m¤¢vebv web¨vm wb¤œiyc :
(i) k Gi gvb wbY©q:
Avgiv Rvwb, m¤¢vebv web¨v‡mi ag©vbymv‡i, P(x) = 1
0.2 + 0.12 + 0.1 + k + 0.25 + 0.15 = 1 k + 0.82 = 1
k = 1- 0.82 k = 0.18
GLb, k Gi gvb ewm‡q ˆ`e PjK x Gi m¤¢vebv web¨vm n‡e wb¤œiyc :
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 k 0.25 0.15
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 k 0.25 0.15
x -3 -1 0 1 2 4
P(x) 0.2 0.12 0.1 0.18 0.25 0.15
(ii) ˆ`e PjK x Gi Mo wbY©q : Mo , E(x) = x P(x)
= (-3)0.2 + (-1)0.12 + 00.1 + 10.18 + 20.25 + 40.15 = - 0.6 - 0.12 + 0 + 0.18 + 0.5 + 0.6 = 0.56
ˆ`e PjK x Gi ‡f`vsK wbY©q :
†f`vsK , V(x) = E (x2) - {E(x)}2 GLv‡b , E(x) = 0.56
Ges E(x2) = x2 P(x)
= (-3)20.2 + (-1)20.12 + 020.1 + 120.18 + 220.25 + 420.15 = 1.8 + 0.12 + 0 + 0.18 + 1 + 2.4 = 5.5
GLb, V(x) = E (x2) - {E(x)}2
= 5.5 - (0.56)2 = 5.5 - 0.3136 = 5.1864
(iii) †`qv Av‡Q, y = 3x + 2
x = -3 n‡j , y = 3(-3) + 2 = -9 + 2 = -7 Ges P(y) = 0.2 x = -1 n‡j , y = 3(-1) + 2 = -3 + 2 = -1 Ges P(y) = 0.12 x = 0 n‡j , y = 3(0) + 2 = 0 + 2 = 2 Ges P(y) = 0.1 x = 1 n‡j , y = 3(1) + 2 = 3 + 2 = 5 Ges P(y) = 0.18 x = 2 n‡j , y = 3(2) + 2 = 6 + 2 = 8 Ges P(y) = 0.25
x = 4 n‡j , y = 3(4) + 2 = 12 + 2 = 14 Ges P(y) = 0.15
ˆ`e PjK y Gi m¤¢vebv web¨vm wb¤œiyc :
(iv) E(y) I V(y) Gi gvb wbY©q : †`qv Av‡Q, y = 3x + 2
E(y) = E(3x + 2)
= 3 E(x) + 2 = 3 0.56 +2 = 1.68 + 2 = 3.68
Avevi, y = 3x + 2
V(y) = V(3x + 2)
= 32 V(x) = 9 5.1864 = 46.6776 mgm¨v (3) : ˆ`e PjK x Gi m¤¢vebv A‡cÿK wb¤œiyc :
P(x) = 4− |5−x|
k , x = 2, 3, 4, … … … , 8
(i) k Gi gvb wbY©q Ki|
(ii) ˆ`e PjK x Gi Mo I †f`vsK wbY©q Ki|
(iii) hw` y = 2x - 3 nq Z‡e E(y) I V(y) Gi gvb wbY©q Ki|
mgvavb : †`qv Av‡Q, ˆ`e PjK x Gi m¤¢vebv A‡cÿK wb¤œiyc : P(x) = 4− |5−x|
k , x = 2, 3, 4, … … … , 8
y -7 -1 2 5 8 14
P(y) 0.2 0.12 0.1 0.18 0.25 0.15
(i) k Gi gvb wbY©q:
Avgiv Rvwb, m¤¢vebv web¨v‡mi ag©vbymv‡i, P(x) = 1 1
k+ 2
k + 3
k + 4
k + 3
k + 2
k + 1
k = 1 16
k = 1
k = 16 k = 16
(ii) GLb, k Gi gvb ewm‡q ˆ`e PjK x Gi m¤¢vebv web¨vm n‡e wb¤œiyc :
ˆ`e PjK x Gi Mo wbY©q : Mo , E(x) = x P(x) = 2 1
16+ 3 2
16+ 4 3
16+ 5 4
16+ 6 3
16+ 7 2
16+ 8 1
16 = 2
16 + 6
16 + 12
16 + 20
16 + 18
16 + 14
16 + 8
16 = 80
16 = 5 ˆ`e PjK x Gi ‡f`vsK wbY©q :
†f`vsK , V(x) = E (x2) - {E(x)}2 GLv‡b , E(x) = 5
Ges E(x2) = x2 P(x) = (2)21
16+(3)22
16+(4)23
16+ (5)24
16+(6)23
16+(7)22
16 +(8)21
16 = 4
16 + 18
16 + 48
16 + 100
16 + 108
16 + 98
16 + 64
16 = 440
16 = 27. 5 GLb, V(x) = E (x2) - {E(x)}2
= 27.5 - (5)2 = 27.5 - 25 = 2.5 (iv) E(y) I V(y) Gi gvb wbY©q :
†`qv Av‡Q, y = 2x - 3
E(y) = E(2x - 3)
= 2 E(x) - 3 = 2 5 - 3 = 10 - 3 = 7
Avevi, y = 2x - 3
V(y) = V(2x - 3)
= 22 V(x) = 4 2.5 = 10
mgm¨v (4) : bxjvi ev‡· 7 wU jvj I 5 wU mv`v ej Av‡Q Ges kxjvi ev‡· 4 wU jvj I 10 wU mv`v ej Av‡Q| bxjv Ges kxjv `yÕRbB ev· n‡Z ˆ`e fv‡e 3 wU ej Zzj‡jv|
(i) bxjvi DVv‡bv ej 3 wUi g‡a¨ mv`v e‡ji msL¨v x aiv n‡j x Gi Mo I †f`vsK wbY©q Ki|
(ii) kxjvi DVv‡bv ej 3 wUi kZ© n‡jv , cÖwZwU mv`v e‡ji Rb¨ 25 UvKv cyi¯‹vi cv‡e wKš‘
cÖwZwU jvj e‡ji Rb¨ 50 UvKv Rwigvbv w`‡Z n‡e| kxjvi jvf n‡e, bv ÿwZ n‡e ?
x 2 3 4 5 6 7 8
P(x) 1
k
2 k
3 k
4 k
3 k
2 k
1 k
x 2 3 4 5 6 7 8
P(x) 1
16
2 16
3 16
4 16
3 16
2 16
1 16
mgvavb : (i) †`Iqv Av‡Q, bxjvi ev‡· jvj ej 7 wU I mv`v ej 5 wU †gvU ej 12 wU
bxjv ev· n‡Z ˆ`e fv‡e 3 wU ej Zzj‡jv| mv`v e‡ji msL¨v x aiv n‡j x Gi gvbmg~n n‡e 0, 1, 2, 3 GLv‡b, P(x = 0) = P( 0 wU mv`v I 3 wU jvj) = 5C0× 7C3
12C3 =1 ×35
220 = 35
220 = 7
44 P(x = 1) = P( 1 wU mv`v I 2 wU jvj) = 5C1× 7C2
12C3 = 5 ×21
220 = 105
220 = 21
44 P(x = 2) = P( 2 wU mv`v I 1 wU jvj) = 5C2× 7C1
12C3 = 10 ×7
220 = 70
220 = 14
44 P(x = 3) = P( 3 wU mv`v I 0 wU jvj) = 5C3× 7C0
12C3 = 10 ×1
220 = 10
220 = 2
44 ˆ`e PjK x Gi m¤¢vebv web¨vm n‡e wb¤œiyc :
* ˆ`e PjK x Gi Mo wbY©q : Mo , E(x) = x P(x) = 0 7
44+ 1 21
44+ 2 14
44+ 3 2
44 = 21
44 + 28
44 + 6
44 = 50
44 = 1.1364
* ˆ`e PjK x Gi ‡f`vsK wbY©q :
†f`vsK , V(x) = E (x2) - {E(x)}2 GLv‡b , E(x) = 1.1364
Ges E(x2) = x2 P(x) = (0)27
44+(1)221
44+(2)214
44+ (3)22
44 = 21
44 + 56
44 + 18
44 = 95
44 = 2.1591 GLb, V(x) = E (x2) - {E(x)}2
= 2.1591- (1.1364)2 = 2.1591 - 1.2914 = 0.8677 (ii) †`Iqv Av‡Q, kxjvi ev‡· jvj ej 4 wU
I mv`v ej 10 wU †gvU ej 14 wU
kxjv ev· n‡Z ˆ`e fv‡e 3 wU ej Zzj‡jv| jvj e‡ji msL¨v 0, 1, 2, 3 n‡j mv`v e‡ji msL¨v 3, 2, 1, 0 n‡e|
awi, cÖvß UvKvi cwigv‡bi PjK x
cÖvß jvj e‡ji Rb¨ Rwigvbv 50 UvKv Ges mv`v e‡ji Rb¨ cyi¯‹vi 25 UvKv GLv‡b, P( 0 wU jvj I 3 wU mv`v) = 4C0× 10C3
14C3 =1 ×120
364 = 120
364 =30
91 G†ÿ‡Î cÖvß UvKvi cwigvb = 0( - 50) + 325 = 0 + 75 = 75 UvKv P( 1 wU jvj I 2 wU mv`v) = 4C1× 10C2
14C3 =4 ×45
364 = 180
364=45
91
x 0 1 2 3
P(x) 7
44
21 44
14 44
2 44
G†ÿ‡Î cÖvß UvKvi cwigvb = 1( - 50) + 225 = - 50 + 50 = 0 UvKv P( 2 wU jvj I 1 wU mv`v) = 4C2× 10C1
14C3 =6 ×10
364 = 60
364=15
91
G†ÿ‡Î cÖvß UvKvi cwigvb = 2( - 50) + 125 = - 50 + 25 = - 25 UvKv P( 3 wU jvj I 0 wU mv`v) = 4C3× 10C0
14C3 =4 ×1
364 = 4
364 = 1
91 G†ÿ‡Î cÖvß UvKvi cwigvb = 3( - 50) + 025 = - 150 + 0 = - 150 UvKv ˆ`e PjK x Gi m¤¢vebv web¨vm wb¤œiyc :
myZivs cÖZ¨vwkZ UvKvi cwigvb , E(x) = x P(x) = 7530
91 + 045
91 + ( 25) 15
91 +( 150) 1
91
= 2250
91 + 0 375
91 150
91 = 1725
91 = 18.96 UvKv
mgm¨v (5) : cvikv GKwU †Mg †Ljvq 2 wU wbi‡cÿ Q°v GK‡Î GKevi wb‡ÿc Ki‡jv|
kZ© n‡jv, hw` †m GKwU Qq †dj‡Z cv‡i Z‡e 15 UvKv jvf Ki‡e Ges `yÕwU Qq †dj‡Z cvi‡j 50 UvKv jvf Ki‡e wKš‘ hw` GKwUI Qq †dj‡Z bv cv‡i Z‡e 8 UvKv Rwigvbv w`‡Z n‡e| cvikvi Kx GB †Mg †Ljv DwPZ n‡e| jvf ÿwZ we‡kølY K‡i gšÍe¨ Ki|
mgvavb : (i) cvikv 2 wU wbi‡cÿ Q°v GK‡Î GKevi wb‡ÿc Ki‡jv|
2wU wbi‡cÿ Q°v GK‡Î GKevi wb‡ÿ‡ci bgybv‡ÿÎwU wb¤œiƒc:
bgybv‡ÿ‡Îi †gvU Dcv`vb msL¨v n(S) = 62 = 36 GKwU‡Z Qq Av‡Q GBi~c bgybvwe›`yi msL¨v 10 wU 15 UvKv jvf Kivi m¤¢vebv 10
36 `yÕwU‡ZB Qq Av‡Q GBi~c bgybvwe›`yi msL¨v 1 wU 50 UvKv jvf Kivi m¤¢vebv 1
36 GKwU‡ZI Qq bvB GBi~c bgybvwe›`yi msL¨v 25 wU 8 UvKv Rwigvbv nIqvi m¤¢vebv 25
36
x 75 0 -25 -150
P(x) 30
91
45 91
15 91
1 91
S 2q Q°v wb‡ÿ‡ci gvb
1 2 3 4 5 6
1g Q°v wb‡ÿ‡ci gvb 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
UvKvi cwigv‡bi PjK x n‡j x n‡j Gi m¤¢vebv web¨vm n‡e wb¤œic:
myZivs cÖZ¨vwkZ UvKvi cwigvb , E(x) = x P(x) = 1510
36 + 501
36 + ( 8) 25
36 = 150
36 + 50
36 200
36 = 0 UvKv †h‡nZz jvf ev ÿwZ †KvbUvB nq bv, myZivs cvikv †Mg †Lj‡Z cv‡i|
4_© Aa¨vq ( wØc`x web¨vm)
## wØc`x web¨vm : †h web¨v‡m ‡Póvi msL¨v mmxg A_©vr 30 Gi †P‡q Kg, †Póv¸‡jv ci¯úi ¯^vaxb, cÖwZevi †Póvq mdjZv I wedjZvi m¤¢vebv aªæeK _v‡K Ges m¤¢vebv؇qi mgwó 1 nq, Z‡e †mB web¨vm‡K wØc`x web¨vm e‡j|
g‡b Kwi, x GKwU wØc`x ˆ`e PjK , wØc`x web¨v‡mi m¤¢vebv A‡cÿKwU wb¤œiƒc:
P(x) = ncx pxqn−x ; x = 0, 1, 2, 3, … … … , n †hLv‡b, n = cixÿ‡bi †Póvi msL¨v
p = cÖwZ †Póvq mdjZvi m¤¢vebv
q = cÖwZ †Póvq wedjZvi m¤¢vebv †hLv‡b, p + q = 1
## wØc`x web¨vmi AšÍwb©wnZ kZ© ev Abygvbmg~n : wØc`x web¨v‡mi AšÍwb©wnZ kZ©mg~n wb¤œiƒc:
(i) cixÿvwU‡Z ‡Póvi msL¨v mmxg A_©vr 30 Gi †P‡q Kg _vK‡e, (ii) †Póv¸‡jv ci¯úi ¯^vaxb n‡e
(iii) cÖwZwU †Póvq mdjZv I wedjZv bv‡g †Kej `ywU djvdj _vK‡e (iv) cÖwZwU †Póvq mdjZv I wedjZvi m¤¢vebv aªæeK _vK‡e
(v) mdjZvi msL¨v x GKwU wew”Qbœ PjK n‡e, †hLv‡b x = 0, 1, 2, 3, … … , n
## wØc`x cixÿv : †h ˆ`e cixÿvq ‡Póvi msL¨v mmxg A_©vr 30 Gi †P‡q Kg, †Póv¸‡jv ci¯úi ¯^vaxb, cÖwZevi †Póvq mdjZv I wedjZvi m¤¢vebv aªæeK _v‡K Ges m¤¢vebv؇qi mgwó 1 nq, Z‡e †mB ˆ`e cixÿv‡K wØc`x cixÿv e‡j|
## wØc`x PjK : †h ˆ`e cixÿvq ‡Póvi msL¨v mmxg A_©vr 30 Gi †P‡q Kg, †Póv¸‡jv ci¯úi ¯^vaxb, cÖwZevi †Póvq mdjZv I wedjZvi m¤¢vebv aªæeK _v‡K Ges m¤¢vebv؇qi mgwó 1 nq, Z‡e †mB ‡ÿ‡Î mdjZvi msL¨v wb‡`©kKvix PjK‡K wØc`x PjK e‡j|
## wØc`x web¨v‡mi ag©mg~n :
(i) wØc`x web¨vm GKwU wew”Qbœ Pj‡Ki web¨vm (ii) wØc`x web¨v‡mi `yBwU civwgwZ h_vµ‡g n I p (iii) wØc`x web¨v‡mi mg¯Í m¤¢vebvi †hvMdj GK|
(iv) cÖwZevi †Póvq mdjZvi m¤¢vebv (p) I wedjZvi m¤¢vebv (q) aªæeK _v‡K|
(v) wØc`x web¨v‡mi Mo n‡jv np A_©vr E(x) = np
(vi) wØc`x web¨v‡mi †f`vsK npq I cwiwgZ e¨eavb √npq (vii) GB web¨vmwUi Mo †f`vsK A‡cÿv eo A_©vr E(x) V(x) (viii) wØc`x web¨v‡mi ew¼gZvsK √ β1 = 𝑞−𝑝
√𝑛𝑝𝑞 ; †hLv‡b p = q n‡j web¨vmwU mylg, p q n‡j web¨vmwU FYvZ¥K ew¼g Ges p q n‡j web¨vmwU abvZ¥K ew¼g
(ix) wØc`x web¨v‡mi m~PjvZvsK β2 = 3 + 1−6𝑝𝑞
𝑛𝑝𝑞
X 15 50 - 8
P(x) 10
36
1 36
25 36
(x) `ywU ¯^vaxb wØc`x Pj‡Ki †hvMdj GKwU wØc`x PjK|
## wØc`x web¨v‡mi †QvU Dccv`¨ :
(i) cÖgvY Ki †h , wØc`x web¨v‡mi Mo †f`vsK A‡cÿv eo A_©vr E(x) V(x) cÖgvY : g‡b Kwi , x GKwU wØc`x PjK hvi civwgwZ n , p †hLv‡b p + q = 1 Avgiv Rvwb, wØc`x web¨v‡mi Mo E(x) = np Ges †f`vsK V(x) = npq GLb, V(x)
E(x) = npq
np = q < 1 A_©vr V(x)
E(x) < 1 V(x) < E(x)
E(x) > V(x)
myZivs, wØc`x web¨v‡mi Mo †f`vsK A‡cÿv eo A_©vr E(x) V(x) (Proved)
(ii) wØc`x web¨v‡mi †cŠbtc~wbK m~Î ev †iKvm©b (Recurssion) m~Î :
†iKvm©b m~Î D™¢veb : g‡b Kwi , x GKwU wØc`x PjK hvi civwgwZ n I p †hLv‡b p + q = 1 Avgiv Rvwb, wØc`x web¨v‡mi m¤¢vebv A‡cÿK
P(x) = ncx px qn−x ; x = 0, 1, 2, 3, … … … , n Ges wØc`x web¨v‡mi †cŠbtc~wbK m~Î ev †iKvm©b (Recurssion) m~Î nj P(x + 1) = n−x
x+1 .p
q . P(x)
(ii) cÖgvb Ki †h,wØc`x web¨v‡mi mg¯Í m¤¢vebvi mgwó GK (1).
cÖgvY: g‡b Kwi , x GKwU wØc`x PjK hvi civwgwZ n I p †hLv‡b p + q = 1 Avgiv Rvwb, wØc`x web¨v‡mi m¤¢vebv A‡cÿK ,
P(x) = ncx px qn−x ; x = 0, 1, 2, 3, … … … , n GLb, mg¯Í m¤¢vebvi mgwó = ∑nx=0P(x)
= ∑nx=0nCx px qn−x
= nc0 p0 qn - 0 + nc1 p1 qn - 1 + nc2 p2 qn - 2 + … ... … + ncn pn qn - n = 1.1. qn + nc1 p1 qn - 1 + nc2 p2 qn - 2 + … ... … + 1. pn .1
= qn + nc1 p1 qn - 1 + nc2 p2 qn - 2 + … ... … + pn = (q + p)n = (1)n = 1 [ q + p = 1 ] wØc`x web¨v‡mi mg¯Í m¤¢vebvi mgwó GK (1). (Proved) MvwYwZK mgm¨vejx :
mgm¨v (1) : GKwU wØc`x Pj‡Ki Mo 4 Ges cwiwgZ e¨eavb √3 (K) ‰`e PjKwUi k~b¨gvb MÖnY bv Kivi m¤¢vebv wbY©q Ki|
(L) P(x ≤ 2) I P(x ≥ 2) Gi g‡a¨ Zzjbv Ki|
(M) web¨vmwUi ew¼gZv I m~uPjZv wbY©q K‡i web¨vmwUi AvK…wZ I cÖK…wZ m¤ú‡K©
gšÍe¨ Ki|
mgvavb : (K) awi, wØc`x PjK x
†`Iqv Av‡Q , Mo E(x) = 4 Ges cwiwgZ e¨eavb √V(x) = √3 ∴ †f`vsK V(x) = 3
Avgiv Rvwb, n I p civwgwZ wewkó wØc`x web¨v‡mi m¤¢vebv A‡cÿK
P(x) = nCx px qn−x ; x = 0, 1 , 2, 3 , … … … , n
wØc`x web¨v‡mi Mo E(x) = np Ges †f`vsK V(x) = npq
∴ np = 4 ... ... ... (i) Ges npq = 3 ... ... ... (ii) (ii) ÷ (i) n‡Z, npq
np = 3
4 ∴ q = 3
4
Ges p = 1- q = 1 - 3
4 = 1
4 (i) n‡Z, n . 1
4 = 4 ∴ n = 16
n , p I q Gi gvb ewm‡q wØc`x web¨v‡mi m¤¢vebv A‡cÿK P(x) = 16Cx (1
4)x ( 3
4)16−x ; x = 0, 1 , 2, 3 , … … … , 16 ˆ`e PjKwUi k~Y¨ gvb MÖnY bv Kivi m¤¢vebv = 1 - P(x = 0)
= 1 - 16C0 (1
4)0 ( 3
4)16−0 = 1 - 1× 1 × 0.0100226 = 0.9899774
(L) P(x ≤ 2) I P(x ≥ 2) Gi g‡a¨ Zzjbv P(x ≤ 2) = P(x =0) + P(x=1) + P(x=2) = 0.0100226 + 16C1 (1
4)1 ( 3
4)16−1 + 16C2 (1
4)2 ( 3
4)16−2 = 0.010023 + 16× 0.25 × 0.01336 + 120 × 0.0625 × 0.01782 = 0.010023 + 0.05344 + 0.13365 = 0.197113
P(x ≥ 2) = 1 - P(x =0) - P(x=1) = 1 - 0.010023 - 0.05344 = 0.84633 ∴ P(x ≤ 2) ≠ P(x ≥ 2)
(M) web¨vmwUi ew¼gZv I m~uPjZv wbY©q : web¨vmwUi ew¼gZvsK √ β1 = q−p
√npq = 0.75−0.25
√16 ×0.25 ×0.75 = 0.5
1.73205 = 0.28868 wØc`x web¨v‡mi m~PjvZvsK β2 = 3 + 1−6pq
npq = 3 + 1−6 × 0.25 ×0.75 16 ×0.25 ×0.75 = 3 + 1−6 × 0.25 ×0.75
3 = 3 + − 0.125
3 = 3 - 0.04167 = 2.95833 †h‡nZz web¨vmwUi √ β1 Gi gvb abvZ¥K Ges β2 ˂ 3
myZivs web¨vmwU abvZ¥K ew¼g Ges AbwZ m~uPv‡jv |
mgm¨v (2) : GKwU KviLvbvi Drcvw`Z `ª‡e¨i 40% Lvivc| `ªe¨¸‡jv 10wUi c¨v‡K‡U mieivn Kiv nq|
(K) †Kvb Lvivc `ªe¨ bv _vKvi m¤¢vebv (L) Kg c‡ÿ 1wU Lvivc `ªe¨ _vKvi m¤¢vebv
(M) eo ‡Rvo 2 wU Lvivc `ªe¨ _vKvi m¤¢vebv wbY©q Ki|
mgvavb : awi, PjK x : Lvivc `ª‡e¨i msL¨v ∴ p = 40% = 0.4
100
40 , q = 1-p =1-0.4 = 0.6 , Ges n = 10
Zvn‡j x GKwU wØc`x PjK|∴ P(x) = ncx px qn−x ; x = 0, 1, 2, 3, … … … , n GLb P(x) = 10cx (0.4)x (0.6)10−x ; x = 0, 1, 2, 3, … … … , 10
(K) †Kvb Lvivc `ªe¨ bv _vKvi m¤¢vebv P(x=0) = 10c0 (0.4)0 (0.6)10−0 = 1× 1×(0.6)10 = .00606
(L) Kg c‡ÿ 1wU Lvivc `ªe¨ _vKvi m¤¢vebv
P(x ≥ 1) = 1 - P(x =0) = 1 - 0.00606 =0.99394 (M) eo ‡Rvo 2 wU Lvivc `ªe¨ _vKvi m¤¢vebv
P(x ≤ 2) = P(x =0) + P(x=1) + P(x=2)
= 0.00606 + 10c1 (0.4)1 (0.6)10−1+10c2 (0.4)2 (0.6)10−2 =0.00606+10 × 0.4×(0.6)9 +45 ×0.16 × (0.6)8
= 0.00606 +0.04031+0.12093 =0.16729