•∫xnemxdx= 1 xnemx− n ∫xn−1emxdx • ∫secnxdx= secn−2xtanx + n−2 ∫secn−2xdx,n≠1 m m n−1 n−1 •∫ emx dx=− emx + m ∫ emx dx,n≠1 • ∫cscn xdx =− cscn−2xcotx + n−2 ∫cscn−2xdx,n≠1 xn (n−1)xn−1 n−1 xn−1 n−1 n−1 •∫sinhnxdx= 1 sinhn−1xcoshx− n−1 ∫sinhn−2xdx • ∫xnlnm xdx = xn+1 lnm x − m ∫xnlnm−1xdx n n n+1 n+1 •∫ dx x =− cosh x − n−2 ∫ dx x ,n≠1 • ∫lnm xdx = lnm x + m lnm−1 xdx ,n≠1 sinhn (n−1)sinhn−1x n−1 sinhn−2 xn (n−1)xn+1 n−1 xn •∫coshnxdx= 1 sinhxcoshn−1xcoshx+ n−1 ∫coshn−2xdx• ∫lnn xdx =x lnnx−n∫lnn−1xdx n n ∫xnsinh xdx = •∫ dx x =− sinh x + n−2 ∫ dx ,n≠1 • coshn (n−1)coshn−1x n−1 coshn−2 x xn cosh x −n∫xn−1cosh xdx ∫xncosh xdx = •∫tanhnxdx=− 1 tanhn−1x+∫tanhn−2xdx,n≠1 • n−1 xn sinh x −n∫xn−1sinh xdx •∫cothnxdx=− 1 cothn−1x+∫cothn−2xdx,n≠1 • ∫xnsin xdx = n−1 •∫sechnxdx= sechn−2xtanhx + n−2 ∫sechn−2xdx,n≠1 −xn cos x +n∫xn−1cosxdx n−1 n−1 ∫xncos xdx = • •∫sinnxdx=− 1 sinn−1xcosx+ n−1 ∫sinn−2xdx n n xn sin x −n∫xn−1sinxdx •∫ dx x =− cos x + n−2 ∫ dx x ,n≠1 • ∫xnsin−1xdx= xn+1 sin−1x− 1 ∫ xn+1 dx sinn (n−1)sinn−1x n−1 sinn−2 n+1 n+1 1−x2 •∫cosnxdx= 1 sinxcosn−1x+ n−1 ∫cosn−2xdx n n xn+1 1 xn+1 dx •∫ dx x = n−1 sin x x + n−2 ∫ dx x ,n≠1 • ∫xncos−1xdx= n+1 cos−1x+ n+1 ∫ 1−x2 cosn cosn−1 n−1 cosn−2 •∫sinmxcosmxdx= sinn+1xcosm−1x + m−1 ∫sinnxcosm−2xdx• ∫xntan−1xdx= xn+1 tan−1x− 1 ∫ xn+1 dx n+m n+m n+1 n+1 1+x2 •∫tann xdx = 1 tann−1 x −∫tanhn−2xdx,n≠1 • ∫ xndx = x − b ∫ dx n−1 axn+b a a axn+b •∫cotn xdx = 1 cothn−1 x −∫cotn−2xdx,n≠1 n−1 • b f(x)dx= nl→im∞ σni=1 f ξi Δxi • b [f(x)+g(x)]dx= b f(x)dx+ b g(x)dx a a a a mxΔxi→0 • ab 1dx=b−a • ab [f(x)−g(x)]dx= ab f(x)dx− ab g(x)dx • b kf(x)dx=k b f(x)dx • ab f(x)dx=0 a a
• ab f(x)dx≥0 if f(x)≥0 on [a,b]. Trapezoidal Rule • b f(x)dx≤0 if f(x)dx+ b f(x)dx for a<c<b • b f x dx= b−a [f x0 +f xn + σin=−11 f xi ] a c a 2n • Fundamental Theorem of Calculus: Simpson’s Rule • ab f(x)dx=F(x)| b =F b −F a if F′ x =f(x) • ab f x dx= b−a [f x0 +4f x1 +2f x2 +4f x3 + a 3n 2f x4 +…+4f xn−1 +f xn ] • Method of Substitution: Area Under a Curve: • ab f(x)dx=cd f(g t )g′(t)dt • S= ab f x dx=F b −F(a) • Integration by Parts: Area Between Two Curves: • S= ab [ f x −g(x)]dx=F b −G(b)−F(a)+G(a) • b udv= uv | b − b vdu a a a • Definite Integral: ab f x dx • f x,y dA= maxl∆imxi→0 m σnj=1 f(ui, vj)∆xi∆yj Continuous Function on [a, ∞): i=1 ∞ f x dx= nl→im∞ n f x dx max ∆yj→0 a a • Continuous Function on (-∞, b]: • Area of a Region: A= b f(x) dydx −b∞ f x dx= n→lim−∞ nb f x dx a g(x) • Volume of a Solid: V= R f x,y dA • ∞ f x dx= c f x dx+ ∞ f x dx • Area and Volume in Polar Coordinates: −∞ −∞ c • Discontinuous Integrand [a, b); discount. @ x=b: • A= s dA = αβ g(θ) rdrdθ; V= s f r,θ rdrdθ h(θ) b f x dx= ε→lim0+ b−ε f x dx ∂z ∂z a a ∂x ∂y • Surface Area:S= R 1+( )2+( )2dxdy • Continuous except for some point c • b f x dx= ε→lim0+ c−ε f x dx+ ε→lim0+ b f x dx • Mass of a Lamina:m= R ρ x,y dA a a c+δ
• Moments of Lamina: • Center of Mass: • Mx= R yρ x,y dA ; My= R xρ x,y dA • x̅ = My = 1 R xρ x,y dA = R xρ x,y dA ; • Moment of Inertia: m m R ρ x,y dA • Ix = R y2ρ x,y dA ; Iy= R x2ρ x,y dA • ȳ= Mx = 1 R yρ x,y dA = R yρ x,y dA • Polar Moment of Inertia m m R ρ x,y dA • Io= R x2+y2 ρ x,y dA • Average of a Function: • μ= 1 R f x,y dA;S= R dA S • G f x,y,z dxdydz= s f x u,v,w ,y u,v,w ,z u,v,w | ∂ x,y,z |dxdydz• Mass of a Solid: m= G μ x,y,z dV ∂ u,v,w • G f x,y,z +g x,y,z dV= G f x,y,z dV+ G g x,y,z dV Center of Mass of a Solid: • • G [f(x,y,z)−g(x,y,z)]dV= G f(x,y,z)dV− G g(x,y,z)dV • G kf x,y,z dV=k G f x,y,z dV • xത= Myz ,ȳ = Mxz ,zlj = Mxy m m m • G∪T f x,y,z dV= G f x,y,z dV+ T f x,y,z dV • Moments of Inertia xy, yz, xz-plane: • G f x,y,z dxdydz= R [ xx12((xx,,yy)) f x,y,z dz]dxdy • Ixy= G z2μ x,y,z dV; • G f x,y,z dxdydz= ab [ cd ( rs f x,y,z dz)dy]dx • Iyz= G x2μ x,y,z dV; • Ixz= G y2μ x,y,z dV; • Volume of a Solid:V= G dxdydz • Moments of Inertia: • Volume in Cylindrical Coordinates: • Ix=Ixy+Ixz= G(z2+y2)μ x,y,z dV • V= S(r,θ,z) rdrdθdz • Iy=Ixy+Iyz= G(z2+x2)μ x,y,z dV • Iz=Ixz+Iyz= G(x2+y2)μ x,y,z dV • Volume in Spherical Coordinates: • Polar Moment of Inertia: • V= S(r,θ,z) r2sinθdrdθdφ • Io=Ixy+Ixz+Iyz= G(x2+z2+y2)μ x,y,z dV
• s F r(s) ds= c F x,y,z ds= c Fds • Mass of a wire: • 0 • m= c ρ x,y,z ds • c1∪c2 Fds= c1 Fds+ c2 Fds • m= c ρ x t ,y t ,z t dx 2 dy 2 dz 2 • dt + dt + dt dt c F x,y ds= aβ F rcosθ,rsinθ dr 2 • r2+ dθ dθ 2 2 + dt dr =τ=(cosα,cosβ,cosγ) m= aβ ρ x t ,y t dx dy ds dt dt • c Pdx+Qdy+Rdz= 0s Pcosα+Qcosβ+Rcosγ ds• Center of Mass of a Wire: Myz Mxy • c Fdr = c1∪c2 Fdr = c1 Fdr + c2 Fdr • xത= m ,ȳ = Mxz ,zlj = m • • m C Pdx+Qdy= aβ df Myz= C xρ(x,y,z)ds, P x,f x +Q x,f x dx dx Mxz= C yρ(x,y,z)ds, • Green′s Theorem: Mxy= C zρ(x,y,z)ds. R ∂Q − ∂P dxdy=∮CPdx+Qdy • Moments of Inertia ∂x ∂y Ix= C y2+z2 p(x,y,z)ds, • Area of a Region R: • Iy= C x2+z2 p(x,y,z)ds, S= R dxdy= 1 ∮Cxdy−ydx Iz= C x2+y2 p(x,y,z)ds. 2 • Path Indepence: • Area of a Region Bounded by a closed Curve: C F(rԦ)⋅drԦ= C Pdx+Qdy+Rdz=u(B)−u(A) S= αβ dy αβ αβ dy • x(t) dt dt=− y(t) dx dt= 1 x(t) dt −y(t) dx dt. dt 2 dt • Test for a Conservative • Volume of a Solid formed by rotating a closed curve: Field: C Pdx+Qdy=u B −u(A) π • V=−π∮Cy2dx=−2π∮Cxydy=− 2 ∮C2xydy+y2dx • Length of a curve: • Work: L= C ds= αβ drԦ dt= αβ dx 2 dy 2 dz 2• W= C Fdr= C Pdx+Qdy dy • dt t dt + dt + dt dt dx dt • W= αβ P(x(t),y(t),z(t)) dt +Q(x(t),y(t),z(t)) + • dt dz Length of a curve in Polar Coordinates: R(x(t),y(t),z(t)) dt • L= αβ dx 2 r 2dθ • Ampere’s Law: ∮CBdr=μ0I dt + • Faraday’s Law: ε=∮CE⋅drԦ=− dψ dt
Order M x, y dx+N x, y dy=0 highest ordered derivative ∂M = ∂N =Exact Degree ∂y ∂x exponent of order General Solution: d2y dy −3x3+2y=8 න M x, y dx + න N x, y dy =C dx2 dx + The equation above is in the 2nd Order Differential Equation and 1st Degree. Standard Linear Form: dx +P x y=Q (x) dy Integrating Factor: v x =e P x dx M x, y dx+N x, y dy=0 General Solution: ;M and N are both functions of x and y. ye P x dx= න Q x e P x dx +c dy =f(x, y)=g x h(y) d xy =xdy+ydx dx General Solution: x ydx−xdy d y = y2 dy =නg x dx+C or H y =G x +C නh y y xdy−ydx d x = x2 d Tan−1 x ydx−xdy y = x2+y2 f λx, λy =λk f (x,y) y = vx or x = vy x dv +v+f v =0 ∂M ≠ ∂N dx ∂y ∂x
d2y +a dy +by=0 Population Growth: dx2 dx P=P0ekt Case 1: Real and Distinct Roots Exponential Growth and Decay: y=C1em1x +C2em2x P=P0ekt Case 2: Real and Repeated Roots Half Life: y=C1em1x +C2xem2x 1 =e−kt 2 Case 3: Conjugate and Complex Roots Newton’s Law of Cooling: y=eax(C1 cos bx+C2 sin bx) T t = T0−Ts e−kt+Ts Case 4: Repeated and Complex Roots Flow Problems: y=eax(C1 cos bx+C2 sin bx) +xeax(C3 cos bx+C4 sin bx) dQ =rate inflow−rate outflow dt Laplace Equation (Elliptic): Heat Equation (Parabolic): Wave Equation (Hyperbolic): ∂2u + ∂2u =0 ∂2u + ∂2u = ∂u ∂2u + ∂2u = ∂2u ∂x2 ∂y2 ∂x2 ∂y2 ∂t ∂x2 ∂y2 ∂t2
an= an−1 + d = an−2 + 2d = … = a1+ (n-1) d an= qan−1= a1qn−1 a1 + an = a2 + an−1 = … = ai + an+1−i a1 . an = a2 . an−1 = … = ai . an+1−i ai−1+ ai+1 ai = ai − 1. ai + 1 2 ai= Sn = anq −a1 = a1(qn−1) q−1 q−1 2a1 + (n−1)d Sn= a1+an . n = 2 . n S = nl→im∞Sn = a1 2 1−q 1 + 2 + 3 + … + n = n(n+1) ∞ 2 an== a1+a2+…an+… Infinite Series: n=1 2 + 4 + 6 + … + 2n = n(n+1) 1 + 3 + 5 + … + (2n-1) = n2 n Sn = an = a1+a2+…an k + (k+1) + (k+2) + … + k(k+n-1) = n(2k+n−1) Nth Partial 2 Sum: n=1 12 + 22 + 32 + … + n2 = n(n+1)(2n+1) 6 13 + 23 + 33 + … + n3 = [ n(n2+1)]2 Convergence ∞ of Infinite an = L if nl→im∞ Sn = L n(4n2−1) Series: n=1 3 12 + 32 + 52 + … + (2n−1)2 = 13 + 33 + 53 + … + (2n−1)3 = n2(2n2- 1) Nth Term Test: ∞ 1 + 1 + 1 + 1 + … + 1 + … = 2 If the series an is 2 4 8 2n n=1 convergent then nl→im∞ an =0. 1 + 1 + 1 +…+ 1 + … = 1 If nl→im∞ an ≠ 0, then the 1·2 2·3 3·4 n(n+1) series is divergent. 1+ 1 + 1 + 1 + … + 1 + … = e 1!· 2!· 3!· (n−1)!
∞∞ Power Series in x ∞ an=A, bn=B (Real Number=c) an xn=a0+a1x+a2x2+anxn+… n=1 n=1 n=1 ∞ ∞∞ Power Series in (x - ∞x0): an (x−x0)n= (an + bn)= an+ bn = A+B n=1 n=1 n=1 n=1 a0+a1 x−x0 +a2(x−x0)2+…+an x−x0 n+… ∞∞ can= c an =cA n=1 n=1 Taylor Series: Interval of Convergence: ∞ ∞ Set of x values in f= an (x−x0)n n=0 f x = f(n) (a) (x−a)n = f(a) + f’(a)(x−a) Radius of Convergence: n! n=0 1 an f\"(a)(x−a)2 fn(a)(x−a)n R= nl→im∞ n an or R= nl→im∞| an+1 | 2! n! + + …+ + Rn Maclaurin Series: f x ∞ (x)n = f(0) + f’(0)x + f\"(0)x2 +… = f(n) =(0) n! 2! n=0 f(n)(0)xn x − 1 1 x − 11)3 1 x − 11)5… n! x + 1 3 x + 5 x + + + Rn ln x = 2 [ + ( + ( ], x > 0 cos x = 1 - x2 + x4 - x6 + … + (−1)nx2n 2! 4! 6! (2n)! ax = 1 + x lna + (x lna)2 + (x lna)3 +…+ (x lna)n + … sin x = x - x3 + x5 - x7 +…+ (−1)nx2n+1 ±… 1! 2! 3! n! 3! 5! 7! (2n + 1)! ln(1 + x) = x - x2 + x3 - x4 + … + (−1)nxn+1 ± …, -1 < x ≤ 1 tan x = x + x3 + 2x5 + 17x7 + 62x9 + … , |x| < π 2 3 4 n+1 3 15 315 2835 2 ln 1 + x = 2 ( x + x3 + x5 + x7 + … ), |x| < 1 cot x = 1 - ( x + x3 + 2x5 + 2x7 + … ) , |x| < π 1 − x 3 5 7 x 3 45 945 4725 arcsin x = x + x3 + 1 ∙ 3x5 + … + 1∙3∙ 5 ... (2n − 1)x2n+1 + ... |x| < 1 2∙3 2 ∙ 4∙5 2∙4 ∙ 6 ... (2n) (2n + 1) arcos x = π - ( x + x3 + 1 ∙ 3x5 + … + 1 ∙ 3 ∙ 5 ... (2n − 1)x2n+1 + … ) , |x| < 1 2 2∙3 2∙4∙5 2 ∙ 4 ∙ 6 ... (2n) (2n + 1)
(1+x)n = 1 + nC1x + nC2x2 + … + mCnxm + … + xn nCm = n (n − 1) ... [n − ( m − 1) ], |x| < 1 a0 ∞ m! 2 f x + ( an cos nx+bn sin nx) n=1 1 = 1 − x + x2 − x3 + ... , |x| < 1 1+x 1 −ππ 1 = 1 +x + x2 + x3 + ... , |x| < 1 an = π f(x) cos nx dx 1+x 1 π 1 − x = x − x2 + 1 ∙ 3x3 - 1 ∙ 3 ∙ 5x4 + … , |x| ≤ 1 bn = π −π f(x) sin nx dx 2 2∙4 2∙4∙6 2∙4∙6∙8 3 1−x = 1+ x − 1∙ ∙x62+13∙ 2∙ 5∙ 9x3-13∙∙26∙∙59∙∙81x24+…,|x| ≤ 1 3 3 ∙6
DESCRIPTIVE STATISTICS σni=1 Range: R= H−L n Mean: xത= xi Standard With frequency distribution table: Deviation: R=(Hmpt −Lmpt ) Median: With frequency distribution table: Mode: xത= σni=1 fixi σin=1 (xi−xത)2 n n−1 S.D.= x=Lm+ n −Fm−1 i With frequency distribution table: 2 fm xො=Lmo+ fmo−f1 S.D.= σin=1 fi(xi−xത)2 2fmo− f1−f2 n−1 NORMAL DISTRIBUTION Quartiles For ungrouped data: Qk= kN f x = 1 e−21 x−σμ 2 4 2πσ2 Normal For grouped data: kN −cfb Distribution: 4 Qk=LB+ i Standard Normal FQ Distribution: z= x−μ Deciles σ For ungrouped data: Dk= kN 10 SAMPLING DISTRIBUTION CONCEPT For grouped data: kN −cfb 10 Dk=LB+ i FQ A Percentiles Proportion of the Population: P= N For ungrouped data: Pk= kN 100 For grouped data: kN −cfb 100 Pk=LB+ i Xഥ−μ FQ σ/ n z=
SAMPLING DISTRIBUTION CONCEPT ESTIMATION With Replacement Large Xഥ−zaൗ2 σ <μ<Xഥ+zaൗ2 σ Sample: n n Total Number of Possible T=Nn Small s s Sample: n n Samples: Xഥ−taൗ2(df) <μ<Xഥ +taൗ2(df) Standard Error: σxഥ = σ n Without Replacement Total Number of Possible T= n! N! ! Samples: N−n a σ N−n Sample p= n n N−1 Proportion: Standard Error: σxഥ = Population p−zaൗ2 p(1−p) <μ<p−zaൗ2 p(1−p) Proportion n n Confidence Interval: Standard Error: σp= P(1−P) Standard Normal Score (z): n z= p−P n= zaൗ22 S2 n= zaൗ22 P(1−P) P 1−P e2 e2 n SAMPLING DISTRIBUTIONS Z-statistic; population standard z= Xഥ−μ z= Xഥ1−Xഥ2 z= Xഥ1−Xഥ2 deviation is known: σ σ12 σ22 s12 s22 n1 n2 n1 n2 Z-statistic; population standard n + + deviation is not known and sample size large (n≥30): z= Xഥ− μ s T-statistic; population standard t= Xഥ1+Xഥ2 deviation is unknown and sample n size is large (n≥30): t= Xഥ−μ ; n1−1 s12+ n2−1 s22 1 + 1 s/ n; n1+ n2−2 n1 n2 with df=n−1 ; with df=n1+ n2−2
SAMPLING DISTRIBUTIONS ANALYSIS OF VARIANCE Single Proportion: z= p−P 1) Compute the various sums of squares (SS): P(1−P) n Sum of Squares n ( σni=1 XT)2 “Total” (SST): NT Two Proportion: z= p1−p2 SST= i=1 XT2 − Sum of Squares p 1−p +p 1−p “Between” (SSB): n 2 σin=1 XT n1 n2 NT 1 Xi ( )2 SSB= ni i=1 − ANALYSIS OF VARIANCE Sum of Squares SSB=SST−SSB “Within” (SSW): 1) Compute the various sums of squares (SS): 2) Partition the between group variance (SSB) Sum of Squares SST= n XT2 − ( σni=1 XT)2 into three independent variances: “Total” (SST): NT • SSBrow (variance for row available) i=1 • SSBcol (variance for column available) Sum of Squares • SSBrow x col (variance due to A & B interaction) “Between” (SSB): n 2 ( σni=1 XT)2 3) Determine the various mean squares (MS): NT dfrow=(r−1); dfcol= c−1 ; dft=(NT−1); SSB= 1 Xi − dfrow x col=(r−1)(c−1); ni i=1 dfw= dft−dfrow−dfcol−dfrow x col Sum of Squares SSB=SST−SSB “Within” (SSW): 4) Determine the various mean: 2) Find the degree of freedom (df): MS rows: MSrow= SSBrow൘dfrow df for “total”: N–1 MS MScol= SSBcol൘dfcol df for “between”: k-1 columns: df for “within”: df for “total” – df for “between” MS rows x MSrow x col= SSBrow x col൘dfrow x col columns: 3) Determine the various mean squares (MS): MS within: MSW= SSWൗdfW MS Between MSB= SSBൗdfB 5) Compute for F-ratio: MS Within MSW= SSWൗdfW Rows: Frow= MSBrowൗMSW Columns: Fcol= MSBcolൗMSW 4) Compute for F-ratio: Cells: Fcol= MSBrow x colൗMSW Fc= MSBൗMSW
REGRESSION ANALYSIS CORRELATION ANALYSIS Yi=α+βXi+εi r= n σ XY − σ X σ Y n σ X2 −( σ X)2 n σ Y2 −( σ Y)2 σ (Xi−Xഥ)(Yi−Yഥ) Formulas for a and b: b= σ (Xi−Xഥ) Alternative Formulas: a=Yഥ−bXഥ b= n σ XY −σXσY n σ X2 −( σ X)2 6 σin=1 Di2 a= σ Y −b σ X ρ=1− n(n2−1) n CHI-SQUARE TEST • σ Yi−Yഥ 2 = σ Yi−Yi 2 + σ Yi−Yഥ 2 χ2= (O−E)2 E SST SSE SSR ; with df=c−1 • SST= σ Yi2 − σ Yi 2 σ Yi2 −nYഥ2 r c (Oij−Eij)2 n = Eij 2 σ XiYi−σ Xinσ Yi 2 χ2= i=1 j=1 σ Xi2− σ Xni 2 • SST= σ Yi2 − σ Yi − ; with df=(r−1)(c−1) n • SSR=SST−SSE χ2= r c (Oij−Eij)2 Eij i=1 j=1 F= MSRൗMSE ; with df=(r−1)(c−1)
Probability of an Event: Permutation: nPm P A = m Combination: nCm n Range of Probability Values: 0 ≤ P(A) ≤ 1 Certain Event: P(A) = 1 n! = (n-2)(n-1)n Impossible Event: 0! = 1 nPn = n! P(A) = 1 nPm = n! Complement: n−m ! P Aഥ =1−P(A) Independent Events: nCm = n = n! P A =P(A) m n−m B m! ! nCm = nCn-m P B =P(B) A nCm + nCm+1 = n+1Cm+1 nC0 + nC1 + nC2 + … + nCn = 2n Addition Rule for Independent Events: P A⋃B =P A +P(B) 1 Multiplication Rule for Independent Events: 11 P A⋂B =P A ∙P(B) 121 1331 General Addition Rule: 14641 P A⋃B =P A +P B −P(A⋂B) 1 5 10 10 5 1 1 6 15 20 15 6 1 P A⋂B =P B ∙ P A = P A ∙P B B A
Law of Total Probability: Standard Z-value: m A Z= X− μ P A = P Bi P Bi σ i=1 Bayes’ Theorem: Cumulative Normal Distribution Function: t−μ 2 A 1 x 2σ2 dt B 2π B P ∙P(B) F x = σ න e A = −∞ P P(A) A P α<X<β =F α−μ −F β−μ Bi σ σ P Bi ∙P P Bi = ε A σkm=1 P Bi A P X−μ <ε =2F σ ∙P Bi Law of Large Numbers: Cumulative Distribution Function: F x =P X<x = −x∞ f t dt P Sn − μ ≥ε →0 as n→∞ n P Sn − μ <ε →1 as n→∞ Bernoulli Trials Process: n μ=np , σ2=npq Chebyshev Inequality: Binomial Distribution Function: P X− μ ≥ε ≤ V(X) b n,p,q = n pkqn−k ε2 k Normal Density Function: μ=np , σ2=npq x−μ 2 f x =(q+pex)n 2σ2 φx= 1 − e σ 2π Geometric Distribution: P T=j = qj−1p, μ= 1 , σ2= q p p2 Standard Normal Density Function: φ z = 1 e−z22 Poisson Distribution: 2π P X=k ≈ λk e−λ, λ=np, μ=λ, σ2= λ k!
Density Function: Markov Inequality: b P(X>k)≤ E(X) P a≤X≤b = න f x dx k a Variance of Discrete Random Variables: Continuous Uniform Density: n 2pi = f= 1 , μ= a+b σ2=V X =E X−μ 2 xi−μ b−a 2 i=1 Exponential Density Function: Variance of Continuous Random Variables: f t =λe−λt , μ=λ, σ2=λ2 ∞ Exponential Distribution Function: σ2=V X =E X−μ 2 = න x−μ 2f x dx F t =1−e−λt −∞ Expected Value of Discrete Random Properties of Variance: Variables: V X+Y =V X +V Y , V X−Y =V X −V Y , n V X+c =V X , μ=E X = xipi V cX =c2V X i=1 Standard Deviation: Expected Value of Continuous Random Variables: D X = V(X)= E[ X−μ 2] ∞ μ=E X = න xf x dx Covariance: cov X,Y =E X−μ X Y−μ Y =E XY −μ(X)μ(Y) −∞ Properties of Expectations: Correlation: E X+Y =E X +E Y , ρ X,Y = cov(X,Y) E X−Y =E X −E Y , V X V(Y) E cX =cE X , E XY =E(X)⋅E(Y) E X2 =V X +μ2
TIME-VALUE RELATIONS Simple Interest: I = Pin F=A (1+i)n−1 F = P(1 + in) i Ordinary Annuity: Number of Interest Periods (n) P=A 1−(1+i)−n i ORDINARY EXACT NORMAL YEAR d/360 d/365 LEAP YEAR d/360 d/366 1−(1+i)−n i Compound Interest: F=P(1+i)n Deferred Annuity: P=A 1+i −m Nominal Rate of Interest: P=F(1+i)-n F=A (1+i)n−1 1+i m i r i = m Effective Rate of ERi = (1 + i)m−1 Annuity Due: P=A 1−(1+i)−(n−1) +1 Interest: i Perpetuity: (1+i)(n−1)−1 F=A i +1 Annuity With Continuous F=P(e)rn Continuous Compounding: Compounding: P= A i Discount: D=F-P Rate of Discount: d=1− 1+i −1 P=A 1−e−rn er−1 ern−1 d= i F=A er−1 1+i (F−P) d= F BOND Annually m=1 Semi-monthly m=24 P=Fr 1+i n−1 +R 1 i 1+i n 1+i n Semi-annually m=2 Weekly m=52 Quarterly m=4 Daily m=365 BREAK-EVEN ANALYSIS Bi-monthly m=6 Continuously m=∞ Total Income = Total Expenses Monthly m=12
TIME-VALUE RELATIONS DEPRECIATION & DEPLETION ARITHMETIC Annual Cost of FC−SV Depreciation: L Total Depreciation d= after n years: Present Worth: PP=AP=AA+1P−G(1+i i)−n Book Value at the end of n years: Dn=nd PG= G 1−(1+i)−n − n Annual Cost of BVn=FC−Dn i i (1+i)n Depreciation: Future Worth: FF=AF=AA+F(1G+i)in−1 Total Depreciation after n years: FG= G (1+i)n−1 −n d=(FC−SV) i i i Book Value at the 1+i L−1 end of n years: Equivalent Uniform A′=A+AG n 1+i n−1 Amount (A’): 1 (1+i)n−1 Annual Cost of i AG=G i − Depreciation: Dn=d Salvage Value: BVn=FC−Dn Total Depreciation GEOMETRIC after n years: n BV L SV Book Value at the k = 1− FC or 1− FC Year 1=A end of n years: Year 2=A(1+f) Year 3=A(1+f)2 Year 4=A(1+f)3 Year n=A(1+f)n−1 d=FC(k) 1−k n−1 CASE 1 CASE 2 SV=FC 1−k L Dn=FC(1− 1−k n) No replacement, maintenance, Replacement only; No and/or operation operation and maintenance CC=FC+P CC=FC+X BVn=FC−Dn ;X= S 1+i k−1
DEPRECIATION & DEPLETION Annual Cost of 2 2 n−1 Sum of the Years n (n+1) Depreciation: L L Digit (SOYD): 2 d=FC( ) 1− SOYD= Salvage Value: Annual Cost of 2 L Depreciation: dn=(FC − SV)( reverse digit ) Total Depreciation L SOYD after n years: SV=FC 1− Book Value at the 2 n Total Depreciation Dn=(FC−SV)( Σ reverse digit ) end of n years: L ) after n years: SOYD Dn=FC(1− 1− BVn=FC−Dn Book Value at the BVn=FC−Dn end of n years: FC = First Cost d = annual cost of depreciation BV = Book Value n = Any Year During the Life of the Property SV = Salvage Value dn = depreciation cost during year n L = Useful Life Dn = Total Cost of Depreciation after n years
DEPRECIATION & DEPLETION Service Method: d = FC−SV Depletion Rate Output Method: hr H Depletion Base −SV dn= d (Hn) = Total Units Expected to be Extracted hr Depletion Charge d = FC−SV = Depletion Rate × Units Extracted During unit T the Period dn= d (Tn) unit SELECTION AMONG ALTERNATIVES Present Worth = PW inflow - PW outflow Excess= Annual Cash inflow – Annual Cash outflow Present Worth Cost = FC + Annual Expenses Annual Cost = Annual Expenses (including depreciation Choose the alternative that has the greatest present and interest on capital) value or least present cost. Choose the alternative that has the greatest excess or least annual cost. Future Worth = FW inflow - FW outflow ROR= Annual Cost A −Annual Cost (B) ቤCapital Invested A −Capital Invested (B)ቤ Future Worth Cost = FC + Annual Expenses If ROR satisfied the minimum required ROR, choose the Choose the alternative that has the greatest future worth alternative with greater investment. Otherwise, choose the or least future worth cost. alternative with less investment or first cost . PAYBACK PERIOD Payback Period years = Investment−SV Net Annual Cash Flow The shorter the payback period, the better the investment.
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