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+ Cx² + Dx+E = 0 x₁ + x₂ + x3 + x4 = -B/A X1X₂ + X1 X3 + + X3 X4 = C/A X1 X2 X3 X1 X2X4+ •+x₂x₂x4 = −D/A X1 X2 X3 X4 = E/A Types of Mean Arithmetic Mean Σχ AM = n Harmonic Mean HM = n/Σ(1/x) Geometric Mean GM=√G₁G₂G3 ... Gn ALGEBRA Quadratic Mean (RMS) QM = √√Ex² /n Arithmetic Progression An = A₁ + (n − 1)d An = Am + (n - m)d Sum of Terms Sn = Geometric Progression Gn = G₁rn-1 Gn = Gmrn-m Sum of Terms Infinite Progression Sn = /[A₁ + A₂] Common Difference d = A₂ - A₁ = An - An-1 Common Ratio SING Soo = [2A₁ + (n − 1)d] Sn = G₁₂ [17²] G₁ 1-r r = Gn Gn-1 Area of a Triangle Right Triangle SSS SAS Inscribed in Circle Circumscribed Circle Escribed Circle A = = ab 2 A = √√s(s − a)(s – b)(s — c) - 1 A = = ab sin C 2 abc 4r A = rs A =r(s-a) A = Where s = 1/²/(a + b + c) PLANE GEOMETRY Median of a Triangle 1 Median = √2(adj² + adj²) — opp² Five-Pointed Star A = 1.123r² Where r = radius of circumscribing circle Circle Area Circumference Arc Length Area of Sector Chord Length Regular Polygon Interior Angle Exterior Angle Diagonals x/2 a α Α = πr2 = Where I = 1 distance from center to chord 0 = central angle in radians R 4 C = 2πγ = πα Arc = re A = r²0 Chord = 2√(r² - 1²) Int L = 180° (n-2) n Ext2 = 360° n No of diagonals = nC2-n a = apothem x = length of one side R = circumradius α = 180°/n Trapezoid Kite A = ² (b₁ + b2₂)h A 1 = 1/2d₁d₂ Parallelogram Given base and altitude A = bh Given diagonals and angle of intersection A = 1/2d₂d₂ sin p Given adjacent side and included angle A = ab sin 0 Rhombus Given one side and altitude A = xh PLANE GEOMETRY Given one side and interior angle A = x² sine Given length of diagonals A 1 == /d₁d₂ General Quadrilateral Bretschneider's Formula A = √(s − a)(s — b)(s — c)(s — d) — abcd cos² 0 - - Where 0 = A+C 2 B+D 2 Bramaguptha's Formula (for cyclic quadrilateral) A = √√(sa)(s — b)(s — c)(s – d) Where r = V (ab+cd)(ac+bd)(ad+bc) 4A Ptolemy's Formula (for cyclic quadrilateral) d₁d₂ = ac + bd Circumscribing a Circle A = √abcd r = A/s Where s = 1/²/(a+b+c+d) Given Diagonals and Angle of Intersection A=d₁d₂ sin o 2 Prism V = Bh SA 2B + Ph = Cylinder V = πr2h SA = : 2π^2 + Ch Cone V = 1²/¬²h 3 SA = π^2 + πrl Pyramid V = Bh 3 SA = B + + Pl SOLID GEOMETRY Frustum V = ²h(B₁ + B₂ + √B₁B₂) 1 SA = B₁ + B₂+ = (P₁ + P₂)l Prismatoid V = 1/(A₁ + 4Am + A₂) Truncated Prism V = A [h₁ + h₂ +h3] 3 Sphere 4 V = SA = 4πr² •πr³ Zone Z = 2πRh Spherical Segment (1 Base) V = n²/² (3R - h) 3 SA = B + Z Spherical Segment (2 Bases) h V = π² (3a²+3b² +h²) SA = B₁ + B₂ + Z Spherical Sector/Cone V = ZR 3 Spherical Wedge V =r³0 0 in radians 3 Spherical Lune SA 2r²00 in radians = Property Faces Vertices Edges Surface Area Volume Radius of Inscribed Sphere Tetrahedron 4 6 12 4AT = √3s² √2 5³ 12 √6 12 S SOLID GEOMETRY Hexahedron 6 8 12 4A, = 6s² S3 ²22 POLYHEDRON Octahedron Dodecahedron Icosahedron 8 12 20 6 Euler's Formula (for any polyhedron that does not intersect itself) F + V - E = 2 12 847 = 2√35² √23 3 √6 6 S 20 30 12 1.11s 30 12A₂ = 20.56s²| 20A = 5√√3s² 7.66s³ 2.18s³ 0.76s Equations of Lines General Equation Two-Point Form Point-Slope Form Intercept Form Slope-Intercept Form Determinant Form Distance Between Points Midpoint Point and Line Parallel Lines d = Ax+By+C =0 32-)n у-У1 X2-X1 X-X1 m = X y + a ANALYTIC GEOMETRY Angle of Inclination tan 0 = m X1+x2 2 = 32-Y1 X2-X1 y = mx + b X y 1- X1 У1 LX2 Y₂ 1] |C₂-C₁1 √A²+B² = 1 d = √√(x₂ − x₁)² + (Y2 − ₁)² xm = Ут = Y₁+Y2 2 d = Ax₁+By₁+C √A²+B² = 0 Triangle by Coordinates Area Basket Method Matrix Method Let Centroid Xc = A== 1₁X₁ X2 X3 X1 2 У1 Уг Уз У1 [X1 Matrix A = y₁ A = 1 X1+x2+x3 3 X2 X3] Уг уз 1 1 1 det (MatA) Yc = Y₁+Y2+Y3 3 Conic Sections General Form Ax² + Bxy + Cy² + Dx + Ey + F = 0 If B = 0, then by the general equation Ax² + Cy² +Dx + Ey + F = 0 e = ANALYTIC GEOMETRY Case 1: If A = 0 or C = 0, then parabola Case 2: If A 1, then circle = C = Case 3: If A# C & same sign, then ellipse Case 4: If A# C & different sign, then hyperbola Eccentricity distance from focus distance from directrix e = 1 0 <e < 1 e > 1 e=0 Parabola Ellipse Hyperbola Circle If B 0, using discriminant B²-4AC = 0 B²-4AC <0 B²-4AC > 0 Circle General Equation x² + y² + Dx+ Ey + F = 0 Center-Radius Form Parabola Circle if A = C Ellipse if A # C Hyperbola (x-h)² + (y-k)² = r² Given Ends of Diameter (x − x₁)(x − x₂) + (y − y₁)(y - y₂) = 0 Ellipse Standard Equation Parallel to x-axis Parallel to y-axis Latus Rectum ade, cea a> b a = de c = ea = √a² - 6² a = semi-major axis b = semi-minor axis LR = (x-h)² (y-k)² a² b² (x-h)² (y-k)² b² a² 26² a ANALYTIC GEOMETRY Hyperbola Standard Equation c = focal distance d = distance from center to directrix e= eccentricity + + = 1 = 1 Parallel to x-axis Parallel to y-axis Latus Rectum a>b, a = de c = ea = √a² + b² a <b, LR= = or a = b a = semi-transverse axis b = semi-conjugate axis Equations of Asymptote (y − k) = ±m(x − h) Where m = a/b m = b/a (x-h)² a² (y-k)² a² 26² a (y-k)² = 1 b² (x-h)² b² = 1 ade, cea if axis is vertical if axis is horizontal Parabola Standard Equation Parallel to y-axis (x - h)² = ±4p(y - k) ANALYTIC GEOMETRY p(+): opens upward p(-): opens downward Parallel to x-axis (y-k)² = ±4p(x - h) p(+): opens to the right p(): opens to the left р = focal distance (focus → vertex → directrix) 2 Area of Parabolic Segment A ==bh 3 Area of Spandrel A = = bh 3 Polar Coordinate System Cartesian to Polar Polar to Cartesian x = r cose y = r sin 0 r² = x² + y² 0 = arctan2 Distance between points X d = r²+r² - 2r₁ r₂ cos(0₂ - 0₁) Trigonometric Functions SOHCAH - TOA Reciprocal Functions CSC X = secx = tan x = 1 sin x sin x COS X cotx = 1 COS X 1 + tan² x = sec² x 1 + cot² x = csc² x COS X sin x Pythagorean Relations sin² x + cos²x = 1 cotx = Sum/Difference of Angles sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y sin x sin y tan(x + y) = tan x+tan y 1-tan x tan y Complementary Angles sin x cos(90° - x) secx = csc(90° - x) cot(90°- x) tan x = 1 tan x TRIGONOMETRY cos x = sin(90° - x) csc x = sec (90° - x) cotx = tan(90° - x) Double Angle sin(2x) = 2 sin x cos x cos(2x) = tan (2x) COS = tan = Half Angle sin () = ± = (cos²x - 2 tan x 1 - tan² x + - sin² x = + 1 - 2 sin² x 2 cos²x - 1 1-cos x 2 |1+cosx 2 1–cosx 1+cos x sin x 1+cos x Cosine Law a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B c²= a² + b² - 2ab cos C 1–cosx sin x Sine Law b sinB a sinA sinc Hyperbolic Function Identities sinh x = tanh x = sech x = ex-e-x 2 sinhx cosh x 1 cosh x cosh x coth x cosh? x – sinh x = 1 1 − tanh x = : sech² x 1 - coth² x = −csch² x cosh(2x) = = csch x = sinh(2x) = 2 sinh x cosh x = exte 2 (cosh? x + sinh x 2 sinh? x + 1 2 cosh² x - 1 cosh x sinhx sinh(x +y) = sinh x cosh y + cosh x sinh y cosh(x + y) = cosh x cosh y + sinh x sinh y 1 sinhx TRIGONOMETRY Rules on Differentiation d -kxn dx d f(x) dx g(x) - kf(x) = kf'(x) dx = d [ƒf(x) ±g(x)] = f'(x) ± g'(x) dx d 1 dx xn ੪ knxn- n-1 dx d (ƒ • g)(x) = f'(g(x))g'(x) dx dx g(x)f'(x)-f(x)g'(x) g² (x) f(x)g(x) = f(x)g'(x) + g(x)f'(x) -n xn+1 1/² √x = 2²/1/Xx dx 2√√x DIFFERENTIAL CALCULUS f(x)9(x) = f(x)^g(x) = [g(x) In f(x)] dx Exponential and Logarithmic Functions d 1 a* = a* In a dx x ln a d dx dex = ex Trigonometric sin x = cos x d dx d dx d dx d dx d dx d dx tan x = sec² x secx = sec x tan x -1 x = tan 1 sec X = d Functions d dx 1 1+x² 1 x√x² -1 Inx nx = -1/ - X dx Inverse Trigonometric Functions 1 -sin-¹ x =√₁²x² 1 ·cos¯¹ x = 1-x d dx d dx loga x = d dx d dx d dx cos x = sin x cotx = -csc² x CSC X= CSC x cotx cot CSC X = X = -1 -1 1+x² -1 x√√x²-1 Hyperbolic Functions d -sinh x = coshx dx d dx -tanh x = d dx d dx csch x = sech² x sech x = - sech x tanh x +∞o +∞o Then dy dx Seven Indeterminate Forms 0 d dx = 818 Parametric Equations Let x = f(t) dx d == f(t) = dt -csch x coth x cosh x dy/dt dx/dt d - coth x = -csch² x dx and y = g(t) dy d dt dt DIFFERENTIAL CALCULUS 0.00 0⁰ 1⁰0⁰ = sinhx d²y dx² g(t) d [dy] d.t dx/dt 8 000 Mean Value Theorem ƒ(b) – ƒ (a) b-a f'(c) = Rolle's Theorem f(x) → [a, b] If f(a) = f(b), then there exists f'(c) = 0 where a < c <b Radius of Curvature [1 + (y')²]³/2 R = ly" | Center of Curvature h = x₁ 1+(y')² -y' 3" Curvature 1 R k y" [1 + (y')²]³/2 k = y₁ + Approximation f(x + Ax) ≈ f(x) + f'(x)Ax 2 1+(y)² DIFFERENTIAL CALCULUS Rate of Change Af(x) Average Rate of Change Ax Instantaneous Rate of Change = f'(x) Ladder Problem L2/3=h²/3 + x²/3 Profit, Cost, Revenue Marginal Profit/Cost = f'(x₁) Actual Profit/Cost = f(x₂) - f (x₁) Revenue = p(x) ·x Where p(x) is the price-demand function Properties of the Integral s dx = x + C S[f(x) ± g(x)]dx = f f(x)dx ± ƒ g(x)dx f Cdx = C f dx Simple Power Formula fxndx = + C xn+1 n+1 Exponential and Logarithmic Functions Sex dx = ex + C ax fax dx + C In a - fx-¹dx = f/dx = ln x + C X Inverse Trigonometric Functions X √ √ ² = xz dx = arcsin + C a INTEGRAL CALCULUS 1 S a²+x² dx = = arctan ²+ C a a 1 1 S x√x²=q² dx = = arcsec² + C a Trigonometric Functions f cos x dx = sin x + C f sin x dx = Stan x dx f cotx dx = In | sin x | + C f secx dx = ln | secx + tan x | + C In | cos x cotx | + C f csc x dx f sec² x dx fcsc² x dx : = f secx tan x dx = secx + C f cos x cotx dx = CSC x + C = - In cos x + C = tan x + C == - cotx + C Hyperbolic Functions [ sinh x dx f cosh x dx = cosh x + C - sinh x+ C f tanh x dx = In | cosh x | + C f coth x dx = ln |sinhx|+C f sech² x dx = tanh x + C fcsch² x dx = - coth x + C f sech x tanh x dx = - sech x + C fcsch x coth x dx = −csch x + C COS X + C Secondary Formulas √ √u² ± a² du = ¹ [u√/u² ± a² ± a² In|u + √u² ± a²|] + C du S √u²³ + a² = In /u + √u² ± a ² | + C và du = uva — u - du 1 S = -In + C u²-a² 2a S du 1 = - In a²-u² 2a + a arcsin (;)] + 1 [u√a² u² C lu-a lu+al ર [u+a] lu-a + C Integration by Parts (LIATE) fudv = uv-fvdu Wallis Formula -π/2 INTEGRAL CALCULUS sinm cos de = (m − 1)!! (n − 1)!! (m + n)!! If m & n are even, then k = π/2 Else, k = 1 k Area Bounded by Curves Vertical Strip •x2 A = [²(Qu-YL) dx Horizontal Strip Y₂ A = [² (XR - XL) dy Jy₁ Polar Coordinates 10₂ L = L = Length of Arc A X1 X1 = ry₂ У1 2 dy [*]++)* 1+ \dx) dx dy r² de 2 dx + 1 dy L = [√²+0)*• L = t₁ 2 .02 [.² √² + (dr) * do de So do dt Centroid Xc = Ус = -X2 x1 ²x(y₁ - y₁)dx Sx²(yu-YL)dx SydA_1² x(vv² - y₁²)dx Jx1 = S dA 2 SX²(VU-YL)dx SxdA S dA x1 X1 Centroid of Hemisphere, d Centroid of Cone, d = h 4 INTEGRAL CALCULUS 4r 4r Centroid of Quarter Circle, (3) 4r Centroid of Semi Circle, (0,7) First Moment of Inertia ry₂ [² Y(XR - XL dy У1 I'x = 4 = x² x(yu-YL)dx l'y X1 Second Moment of Inertia [²y² (XR - XL)dy Jy₁ -X2 I'" = 1y = x² (yu-YL)dx X1 Find 2nd moment unless stated otherwise Radius of Gyration 12 A A Find magnitude of radius unless stated otherwise rx = ry = r = 2 rx² 2 +ry² Volume Generated 1) Washer/Disk Method (1 to axis) •x₂ [*²* (v ₁³² - y₁2² ) dx X1 V = π FS ² (XR² - X₁²) dy Y1 V = π 2) Cylindrical Shell Method (II to axis) -X2 V = 2π ² x(yu-YL) dx X1 [²Y(XR-X₁) dy V = 2π INTEGRAL CALCULUS Pappus's Theorem 1) First Proposition У1 S=CL = 2πrL Where S = surface area C = circumference of curve r= distance of centroid to axis of rotation L = length of curve 2) Second Proposition V = CA = 2πrА Where V = volume generated A = area bounded Methods of Solution for First Order DE A. Variable Separable Standard Form General Solution B. Homogeneous of the Same Degree Standard Form General Solution OF Steps in solving 1. Replace x with vy or y with vx (Note: Use differential multiplied to a smaller number of terms) 2. Simplify by dividing by a common factor and/or cancelling out terms 3. Variable separable method may be applied 4. Replace v using x = vy or y = vx C. Exact Differential Equations Standard Form General Solution Condition for Exactness OF ду əx = DIFFERENTIAL EQUATIONS M(x) dx + N(y)dy = 0 S M(x) dx + f N(y)dy = C M(x, y) M(x,y)dx + N(x, y)dy = 0 f(x, y) = C = = N(x, y) M(x,y) dx + N(x, y)dy f(x, y) = C ƏM ду ƏN əx D. Linear Differential Equations Standard Form +yP(x) = Q(x) General Solution yeſP(x)dx = [ Q(x)eſ P(x)dxdx + C dy dx E. Determination of Integrating Factor Given a differential equation M(x, y)dx + N(x,y)dy which is not exact Let u = f(x, y) be an integrating factor Case 1. If u = f(x), u=e³ Case 2. If u = f(y), u = e = (2x ƏM ƏN Sax əx ƏN ƏM əx əy U = Case 3. For homogeneous functions Mx + Ny # 0 1 Mx+Ny Case 4. Equations in the form U = dy f₁(x, y)ydx + f₂(x, y)xdy = 0 with Mx - Ny # 0 1 Mx-Ny F. Bernoulli Differential Equation Standard Form (in y) +yP(x) = y¹Q(x); n ‡ 0 dy dx General Solution y(1-n)e(1-n) ƒ P(x)dx = (1 − n) ſ Q(x)e(1-n) ƒ P(x)dx dx + C G. Ricatti Differential Equation a(x)y² + b(x)y + c(x) y' DIFFERENTIAL EQUATIONS = Higher Order Differential Equations Case 1. Roots are real and distinct y = C₁e¹₁x + C₂em ₂x + С3em3x Case 2. Roots are real and repeated y = C₁emx + C₂xemx + C3x² emx Case 3. Roots are complex and distinct Root m = a + bi y = eax (C₁ cos bx + C₂ sin bx) Case 4. Roots are complex and repeated y = eax (C₁ cos bx + C₂ sin bx) + xe ax (C3 cos bx + C4 sin bx) Euler-Cauchy Higher Order DE d²y dy dx² dx ax². + bx. + cy = 0 Substituting y = xm, it follows that am(m1) + bm + c = 0 Case 1. Roots are real and distinct y = C₁xm₁ + C₂xm² + С3 xm3 Case 2. Roots are real and repeated y = C₁xm + C₂xm ln x + C3xm ln² x Case 3. Roots are complex and distinct y = xa [C₁ sin(b ln x) + C₂ cos(b ln x)] Case 4. Roots are complex and repeated y = xª [C₁ sin(b ln x) + C₂ cos(b ln x)] +xª lnx [C3 sin(b ln x) + C cos(b ln x)] Applications of Ordinary Differential Equations 1. Growth and Decay dQ dt Q = Cekt 2. Newton's Law of Heating/Cooling dT xT-Tr T-Tr= Cekt dt x Q 3. Rectilinear Motion Vf = V₁ + at S-S₁ = v₁t + 1²/1at² 2 v²-v₁² S - So = 2a 4. Dilution Problem dQ dt Co DIFFERENTIAL EQUATIONS = R¡C¡ – RoCo Q V₂ = V = V + (R₁ − Ro)t 5. Fourier's Law of Heat Conduction dT Q = -KA dx 6. Flow of Fluid through Orifice dV So dt z = depth of fluid at any time t = area of the hole = -kSo√2gz = g = acceleration due to gravity 0.6 (sharp edged) 7. Dissolution dQ dt = k(A - Q)(Cs - Ci) Q = amount of substance dissolved A = amount of substance supplied Cs = saturation concentration C₁ = instantaneous concentration 8. Orthogonal Trajectory dx (d²)₂ = (-dy) c dy T Definition of Laplace Transform ∞ F(s) = L{f(t)} = "* est f(t)dt 0 Laplace Transform of Common Functions L{k} = k/ S L{t} = 1/1/23 s² L{t²} = L{t”}: = L{eat} = O 2! S3 n! sn+1 1 s-a Important Note O DIFFERENTIAL sinh x = L{e-at} = L{sin wt} 1 s+a = L{cos wt}: = _{sinh at} W s²+w² = S s²+w² L{cosh at}= = a s²-a² S s²-a² n in the should be a positive integer The wt in the sine and cosine functions is in radians Recall the definition of hyperbolic functions ex - e ex + e-x 2 2 EQUATIONS Theorems on Laplace Transform 1. Linearity of Laplace Transform If L{f(t)} = F(s) and L{g(t)} = G(s), a and b are constants, then L{af (t) + bg(t)} = aF(s) + bG(s) cosh x = 2. S-Shifting Theory If L{f(t)} = F(s), then L{eat f(t)} = F(s − a) 3. T-Shifting Theorem If L{ƒ (t)} = F (s), then L{u(t – a)ƒ(t − a)} = e¯ª³F(s) Laplace Transform of Derivatives If L{y(t)} = Y(s), then L{y' (t)} = sY(s) — y(0) L{y"(t)} = s²Y(s) - sy(0) - y'(0) L{y""(t)} = s³Y(s) - s²y(0) - sy'(0) - y" (0) Measures of Central Tendency Mean average Median - middle value Mode - highest frequency O Types of Mean Arithmetic Mean Σχ n AM = Weighted Average Σfx WM = f Geometric Mean GM="G₁G₂G3 ... Gn Harmonic Mean HM = Quadratic Mean (RMS) |Σχ2 n QM STATISTICS AND PROBABILITY Measures of Dispersion o Range = Max - Min n Σ(1/x) = O O Max+Min 2 Midrange = Standard Deviation Sample, Sx = Σ(x-x)² Population, ox = n Note: Sample SD is the default Variance 2 = Sample, sx |Σ(x-x)2 n-1 Σ(x-x)² n-1 2 Σ(x-x)² Population, ox² = n O Mean Absolute Deviation, MAD = O Interquartile Range = Q3 - Q₁ Semi-interquartile Range Q3-Q₁ 2 = Σ|x-x| n Grouped Data Mean, x = Median Σ(CM·f) Ef CM = class mark (middle value of class) f = frequency of class = Ime + N 2 -CFme-1 fme Mode = lmo + STATISTICS AND PROBABILITY Ime = lower boundary of median class N = total number of data W CFme-1 = cumulative frequency before median class fme = frequency of median class w = class width fmo-1-fmo fmo-1+fmo+1 1 = lower boundary of modal class fmo = frequency of modal class fmo-1 = frequency before modal class fmo+1 = frequency after modal class Hypothesis Testing O O O Type I Error - false positive; accept Ho Type II Error - false negative; reject Ho If p - value ≥ a, then fail to reject Ho If p-value <a, then reject Ho Confidence Level Reject Ho Critical Value 8|N a Rejection region C = 1-a Do not reject Ho 1-α Nonrejection region Reject Ho Critical Value a Rejection region Permutation n! (n −r)! Permutation of Distinct Objects nPr = n! a! b! c! Circular Permutation P = (n − 1)! Combination nCr Circular Permutation in Space (n − 1)! P = 2 where a+b+c=n = STATISTICS AND PROBABILITY Hockey Stick Identity rCr +r+1Cr +r+₂Cr + + nCr = n+1Cr+1 n! (n − r)!r! Stars and Bars Method For positive integers For non-negative integers n Pr r! s-1Cr-1 s+r-1Cr-1 Chicken McNuggets Theorem n = ab-a-b Derangement n ! Σ (-1) * x! x=0 !n = n! Probability Theory Mutually Exclusive Events P = P₁ + P₂ Dependent/Independent Events P = P₁P₂ Principle of Inclusion and Exclusion P(A n B) = P(A) + P(B) − P(A U B) Conditional Probability P(An B) P(B|A) P(A) = Binomial Distribution P = nCr pr. qn-r Where n = number of trials r = number of successful trials p = probability of success q=probability of failure Multinomial Distribution P = STATISTICS AND PROBABILITY n! a! b!c! Where P₁ + P₂ + P3 = 1 and a+b+c=n - (P₁) ª (P₂) (P3) Hypergeometric Distribution P mCanCb m+na+b Normal Distribution Xs X Z = P(x < xs) = P(z) P(x > xs) = R(z) P(xs < x < x) = Q(z) O z-Score of Sample Mean xs-x o/√n Z= Poisson Distribution e-^.^n n! P = Mathematical Expectation E = Σp • W – Σq · L Odds Odds happening = P 1-P Odds not happening = 1-P P