ze`gqep s — dwihqihhq dwixehpianewx ql zeiexyt`d xtqn •

n! dxeya mipey mixai` n –

(n − 1)! lbrna mipey mixai` n –

n!n1!···nk! i beqn mixai` ni mpyi xy`k dxeya mibeq k-n mixai` n –:minb n •xe q `l xe q mb n

(

nk

)

= n!k!(n−k)! (n)k = n!

(n−k)! dxfgd `ll(k+n−1

k

)

= (k+n−1)!k!(n−1)! nk dxfgd mr

(nk

)

=(n−1k−1

)

+(n−1

k

) lwqt yleyn zgqep •

(a + b)n =∑n

i=0

(

ni

)

aibn−i oeheip ly mepiad •∑k

i=0

(ni

)( mk−i

)

=(n+m

k

)

Vandermonde ly zedfd •zexazqd iagxn:zexazqd ziivwpet zepekz •.A ⊆ Ω rxe`n lkl P (A) ≥ 0 .1.P (Ω) = 1 .2.P (∪iAi) =∑

i P (Ai) f` zebefa mixf zerxe`n A1, A2, . . . m` .3:miixhniq zexazqd iagxn •P (A) = |A|/|Ω| miiwzn (Ω, P ) ixhniq i a zexazqd agxna A rxe`n lklP (A ∩ B) = P (B|A)P (A) :zipzen zexazqd •P (∩n

i=1Ai) = P (A1)P (A2|A1) · · ·P (An| ∩n−1i=1 Ai) :ltkd zgqep •

∪iAi = Ω-y jk zebefa mixf zerxe`n ly dx q A1, A2, . . . m` :dnlyd zexazqdd zgqep •P (B) =

∑

i P (B|Ai) · P (Ai) f`P (C|B) = P (B|C)·P (C)

P (B) :qiia htyn •

P (A ∩ B) = P (A)P (B) m` miielz izla md B-e A zerxe`n •:miiwzn I ⊆ 1, . . . , n miqw pi` ly dveaw lkl m` miielz izla md A1, . . . , An zerxe`n •P (∩i∈IAi) =

∏

i∈I P (Ai) miixwn mipzyn:m` X i a ixwn dpzyn ly zexazqd ziivwpet `id PX •

x lkl PX(x) ≥ 0 –

∑

x PX(x) = 1 –:m` (X,Y ) i a ixwn xehwe ly zexazqd ziivwpet `id PXY •

y-e x lkl PXY (x, y) ≥ 0 –

∑

x

∑

y PXY (x, y) = 1 –:zeiley zeiebltzd •

PX(x) =∑

y PXY (x, y) –

PY (y) =∑

x PXY (x, y) –

PXY (x, y) = PX(x) · PY (y) miiwzn y-e x lkl m` miielz izla md Y -e X •3

zlgezE(X) =

∑

x x · P (X = x) zlgez •E(f(X)) =

∑

x f(x) · P (X = x) n''n ly divwpet ly zlgez •E(f(X,Y )) =

∑

x

∑

y f(x, y) · P (X = x, Y = y) n''n ipy ly divwpet ly zlgez •:zlgez ly zepekz •

E(C) = C C reaw ly zlgez .1E(

∑ni=1 αiXi) =

∑ni=1 αiE(Xi) zlgezd ly zeix`pil .2

E(X) =∑∞

k=1 P (X ≥ k) f` ,ilily i` n''n X idi •E(XY ) = E(X)E(Y ) m` min`ezn izla mi`xwp Y -e X •zipzen zlgezE(X|A) =

∑

x x · P (X = x|A) :A rxe`n ozpda zipzen zlgez •f` ∪iAi = Ω-y jk zebefa mixf zerxe`n ly dx q A1, A2, . . . m` :dnlyd zlgezd zgqep •E(X) =

∑

i E(X|Ai) · P (Ai) zepeyVar(X) = E((X − µ)2) = E(X2) − E(X)2 zepey •σ(X) =

√

Var(X) owz ziihq •:zepey ly zepekz •.P (X = C) = 1 m''m` Var(X) = 0 ,ok enk .X n''n lkl Var(X) ≥ 0 .1Var(aX + b) = a2Var(X) b-e a mireaw ipy lkl .2Var(

∑ni=1 Xi) =

∑ni=1 Var(Xi) f` zebefa min`ezn izla X1, . . . ,Xn m` .3m`zn m wne ztzeyn zepey

Cov(X,Y ) = E[(X − µX)(Y − µY )] = E(XY ) − E(X)E(Y ) :ztzeyn zepey •:ztzeyn zepey ly zepekz •

Cov(X,Y ) = Cov(Y,X) zeixhniq .1Cov(aX + b, cY + d) = ac · Cov(X,Y ) .2Cov(X,Y + Z) = Cov(X,Y ) + Cov(X,Z) .3Var(

∑

i Xi) =∑

i Var(Xi) +∑

i6=j Cov(Xi,Xj) miixwn mipzyn ly mekq ly zepey •

X = X−µX

σXowezn ixwn dpzyn •

ρ(X,Y ) = Cov(X,Y )σ(X)σ(Y ) = Cov(X, Y ) m`zn m wn •mitivx miixwn mipzyn.x ∈ R lkl P (X = x) = 0 m` sivx `xwi X ixwn dpzyn •

FX(x) = P (X ≤ x) sivx n''n ly zxahvnd zebltzdd ziivwpet •miiwzn x ∈ R lkly jk fX zilily i` diivwpet `id X sivx n''n ly zetitvd ziivwpet •FX(x) =

∫ x−∞ fX(t)dt

E(X) =∫ ∞−∞ tfX(t)dt sivx ixwn dpzyn ly zlgez •

E(g(X)) =∫ ∞−∞ g(t)fX(t)dt sivx ixwn dpzyn ly divwpet ly zlgez •

Var(X) =∫ ∞−∞(t − µ)2fX(t)dt = E(X2) − E(X)2 sivx ixwn dpzyn ly zepey •

FXY (x, y) = P (X ≤ x, Y ≤ y) mitivx n''n ipy ly ztzeynd zebltzdd ziivwpet •FY (y) = FXY (∞, y) ,FX(x) = FXY (x,∞) zeiley zebltzd zeivwpet •FXY (x, y) = FX(x) · FY (y) m''m` miielz izla md Y -e X •fXY (x, y) = fX(x) · fY (y) m''m` miielz izla md Y -e X •4

ihx phq ilnxep ixwn dpzynX = X−µ

σ ∼ N(0, 1) f` X ∼ (µ, σ2) m` •FZ(z) = Φ(z) `f Z ∼ N(0, 1) m` •

Φ(−z) = 1 − Φ(z) •leab ihtyne mipeieeiy i`P (X ≥ a) ≤ E(X)

a miiwzn a > 0 lkl f` ilily i` n''n `ed X m` :aewxn oeieeiy i` •t > 0 lkl f` zeiteq Var(X) zepeye E(X) zlgez lra n''n `ed X m` :ayia'v oeieeiy i` •

P (|X − E(X)| ≥ t) ≤ Var(X)t2

miiwznly ylgd wegdy xn`p .µi = E(Xi) zeiteq zelgez ilra n''n ly dx q X1,X2, . . . idz •limn→∞ P (|Xn − µn| ≤ ε) = 1 miiwzn ε > 0 lkl m` dx qd lr lg mile bd mixtqnd

µn = 1n

∑ni=1 µi-e Xn = 1

n

∑ni=1 Xi xy`k

Var(Xi) = σ2-e E(Xi) = µ miiwzny jk miielz izla n''n ly dx q `id X1,X2, . . . m` •.dx qd lr lg mile bd mixtqnd ly ylgd wegd f` ,i lkllimn→∞ Var(Xn) = 0 :mile bd mixtqnd ly ylgd wegd meiwl witqn i`pz •zepeye µ zlgez ilra ,zebltzd ieeye miielz izla n''n X1,X2, . . . eidi :ifkxnd leabd htyn •

limn→∞ P ( Xn−µσ/

√n≤ x) = Φ(x) miiwzn x ∈ R lkl f` ,zeiteq σ2:dpwqn

Xn ; N(µ, σ2/n) –

∑ni=1 Xi ; N(nµ, nσ2) –

Var(X) E(X) PX(k) mikxr ixwn dpzyn(N−M+1)2−1

12M+N

21

N−M+1 M, . . . ,N X ∼ U(M,N) ig`p(1 − p) p pk(1 − p)1−k 0, 1 X ∼ Ber(p) ilepxanp(1 − p) np

(nk

)

pk(1 − p)n−k 0, . . . , n X ∼ Bin(n, p) inepian(D

N )(N−DN )(N−n

N−1 ) nDN

(Dk

)(N−Dn−k

)

/(N

n

)

0, . . . , n X ∼ HG(n,N,D) ixhne`ibxtid1−pp2

1p (1 − p)k−1p 1, 2, . . . X ∼ G(p) ixhne`ib

λ λ λk

k! e−λ 0, 1, . . . X ∼ Pois(λ) ipeq`et

Var(X) E(X) fX(x) mikxr ixwn dpzyn(b−a)2

12a+b2

1b−a [a, b] X ∼ U [a, b] ig`

1λ2

1λ λe−λx

R≥0 X ∼ Exp(λ) ikixrn

σ2 µ 1√2πσ2

e−(x−µ)2

2σ2 R X ∼ N(µ, σ2) ilnxep1 0 1√

2πe−

x2

2 R Z ∼ N(0, 1) ihx phq ilnxep

5