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6 6 R 3 B > / ?429 Maximizing probability of satisfying a linear inequality Let c be a random variable in Rn, normally distributed with mean ¯c and covariance matrix R Consider the problem maximize prob(cTx ≥ α) subject to Fx g, Ax = b Find the conditions under which this is equivalent to a convex or quasiconvex optimizaY = C I G X – M in equilibrium Y = 125 075(Y100) 1 100 = 345 075Y – 75 Y = 270 075Y 025Y = 270 Y = 1080 (b) C = Consumption function = 075(Y – T) T = 02Y G = Government Spending = 50 I = Investment Spending = X = M 10 Y = C I G X – M in equilibrium Y = 075(Y – 02Y) 50 10 = 100

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'tŠy ƒCƒ‰ƒXƒg ‚©‚í‚¢‚¢-C Mdx Ndy Parametrize the curve x= t, y= t, with tfrom 0 to 1 Put everything in terms of t dx= dt dy= dt M= x2y= t3 N= x 2y= t Now we put this into the integral I= Z C Mdx Ndy= Z 1 0 t3 dt tdt= Z 1 0 t3 tdt= 1 4 This is a di erent value from Example GT3, which leads to the important principle Important principle for line integralsY = C I G (XM) National income with government interference From the national income, some will be used for consumption, saving, and other's used for paying taxes With adding of variable taxes on national income, then the formula is Y = C S T To see the balance of national income with the adding of government spending and tax

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5 (Logan, 24 # 1) Solve the problem ut =kuxx, x >0, t >0, ux(0,t)=0, t >0, u(x,0)=φ(x), x >0, with an insulated boundary condition by extending φ to all of the real axis as an even function The solution is u(x,t)= Z ∞ 0 G(x −y,t)G(x y,t)φ(y)dy First note that the solution to the IVP ut = kuxx, −∞ < x < ∞, t > 0, u(x,0) = f(x), −∞E x p a n d i n g s u p p o r t b e y o n d t h e v i r t u a l c l a s s r o o m M an d y S av i tz Romer i s th e N an cy Pf or z h ei mer A r on son S en i or Lectu r er i n H u man Dev el op men t an d Ed u cati on at th e H ar v ar d Gr ad u ate S ch ool of Ed u cati on(Y − T TR) is disposable income whereas (Y − T TR − C) is private saving Public saving, also known as the budget surplus, is the term (T − G − TR), which is government revenue through taxes, minus government expenditures on goods and services, minus transfers Thus we have that private plus public saving equals investment

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Substituting 2T for T in X(w), we have Y(w) = 2(2T) sin(w2T) The zero crossings are at w,2T = nr, or w = n S96 (a) x(t) = Substituting t X(w)et dw = 0 in the preceding equation, we get x(0) = 1 27r X( ) de (b) X(w) = f x(t)e jwt dt Substituting o = 0 in the preceding equation, we getN O A A ' S E C O L O G I C A L F O R E C A S T I N G H ypo xia Fo recas t s H y p o x i a ( l o w o x y g e n ) h a s d r a m a t i c a l l y i n c r e a s e d i n ULu= g(x;y) (111) is called inhomogeneous linear equation Notice that if uh is a solution to the homogeneous equation (19), and upis a particular solution to the inhomogeneous equation (111), then uhupis also a solution to the inhomogeneous equation (111) Indeed

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 ~ i p T u U s ¿ T g T ` c ~ l T i U x l Y S U p c T Y g h x i M g ` " Y U T Y l S i T Y r i T1 Show that if X and Y are independent random variables with the moment generating functions M X(t) and M Y (t), then Z = X Y has the moment generating function, M Z(t) = M X(t)M Y (t) 2 Find a variance of the random variables in Example 1 Finally, we can also define the conditional expectation, E(X Y), and conditional variance, E(X I n t e g r i t y S e r v i c e E x c e l l e n c e Ramstein Senior Officer Quarters Basement STORAGE 15 6'2"X11'3"

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Two Examples of Linear Transformations (1) Diagonal Matrices A diagonal matrix is a matrix of the form D= 2 6 6 6 4 d 1 0 0 0 d 2 0 0 0 0 d n 3 7 7 7 5 The linear transformation de ned by Dhas the following e ect Vectors areC depends on disposable income, not Y We have been able to ignore this difference up to now, since Tx has been zero until now We solve in exactly the same way, but remember to substitute for DI = YTx Definition AE = CIGXIM In this case, substituting in for the variables AE = a 0 MPC*YTx I 0 G 0 X 0IM 0 = 75YP t c a u s l r e t 1 2 / 3 0 W h a t C u s e D m

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* , , / 0 1 ( 2 ) '!Uxy = f′(x)g′(y) Substituting into the PDE, we have uuxy = f(x)g(y)f′(x)g′(y) = uxuy Hence, u(x,y) = f(x)g(y) is a solution of the PDE 3 Boundary value problem The Poisson's Equation is the nonhomogeneous version of Laplace's Equation ∂2u ∂x2 ∂2u ∂y2 = ρ(x,y) (1) Assume that ρ(x,y) = 1 (a) Find the condition underÇ ¶ t ù d o Í É ç w M ü Z U D ó t s b { Í É ç ¨ U ¨ T ¬ b ³ æ ¶ ¬ ¶ 7 G Z X ¹ å Í É ç Z ³ æ ¶ ' ± ¶ Ü 451 4 8GIN ¬ ¶ 7 G Z 8 150cm 4 ± ¶

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 Definition Standard macroeconomic theory points to how a budget deficit can be a contributing factor to a current account deficit This link can be seen from considering the national accounting model of the economy = (), where Y represents national income or GDP, C is consumption, I is investment, G is government spending and X–M stands for net exportsG(x) = sup y∈A f(x,y) is convex examples • support function of a set C SC(x) = supy∈C yTx is convex • distance to farthest point in a set C f(x) = sup y∈C kx−yk • maximum eigenvalue of symmetric matrix for X ∈ Sn, λmax(X) = sup kyk2=1 yTXy Convex functions 3–169 1 / ;

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BASIC STATISTICS 5 VarX= σ2 X = EX 2 − (EX)2 = EX2 − µ2 X (22) ⇒ EX2 = σ2 X − µ 2 X 24 Unbiased Statistics We say that a statistic T(X)is an unbiased statistic for the parameter θ of theunderlying probabilitydistributionifET(X)=θGiventhisdefinition,X¯ isanunbiasedstatistic for µ,and S2 is an unbiased statisticfor σ2 in a random sample 3Partial derivatives of composite functions of the forms z = F (g(x,y)) can be found directly with the Chain Rule for one variable, as is illustrated in the following three examples Example 1 Find the xand yderivatives of z = (x2y3 sinx)10Idaho Fishing Seasons & Rules 1921 idfgidahogov 31 Southeast Special Rule Waters for the Southeast Region For waters not listed below, see page 30 for fishing seasons, bag and possession limits, and rules that apply to all waters within the Southeast Region

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The reason behind this is that the definition of the mgf of X Y is the expectation of et(XY ), which is equal to the product etX ·etY In case of indepedence, the expectation of that product is the product of the expectations • While for independent rv's, covariance and correlation are always 0, the converse is not true OneO x f d H b k w o x f r d h a n b k s c m / v i e 1 0 9 3 7 8 5 6 2 4? The parameterization \(x = h\left( t \right)\), \(y = g\left( t \right)\) will then determine an orientation for the curve where the positive direction is the direction that is traced out as \(t\) increases Finally, let \( C\) be the curve with the same points as \(C\), however in this case the curve has \(B\) as the initial point and \(A

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Drones asteroids Aerial Analysis – Challenge 1 Some humans see a photo as an image that perhaps captures a moment in time To thinking men, it can be carefully read to see what has happened in the past and perhaps what might occur in the futureX Y L8 ¨ Z X Y X Y L8 '4 4 61 ³ æ ¶ w Ô ù Ø u 7 G Æ ;Words that Start with x Words that Start with y Words that Start with z Unscrambled Words That End With s ed ing en s en er est ian ize ise fy ly ful able ible hood ness less ism ment ist al ish Typing Word Game Click "Play Now" to Start!

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Or if and only if there exists m>0 such that r2f(x) mI;% 3 445 " % & 3 " $ 3 44 $ $ 3 " 6 7 8 5" & !X(t) = E etX If M X(t) = M Y(t) for all tin an interval around 0 then X =d Y The moment generating function can be used to \generate" all the moments of a distribution, ie we can take derivatives of the mgf with respect to tand evaluate at t= 0, ie we have that M(n) X (t)j t=0 = E(X n) 3

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Consider now two random variables X,Y jointly distributed according to the pmf p(x,y) We now define the following two quantities Definition The joint entropy is given by H(X,Y) = − X x,y p(x,y)logp(x,y) (4) The joint entropy measures how much uncertainty there is in the two random variables X and Y taken together/0 $1 ' !1 " * 2 "!

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Y =NominalGDP Y ≡C I G (X −M) 7 Consumption • Durable Goods 3 years rule • Nondurable Goods • Services 8 Gross Private Investment (I) • Nonresidential Fixed Investment • Residential Fixed Investment • Inventory Investment • Stocks vs Flows 98x2dom(f) One of the main uses of strict convexity is to ensure uniqueness of the optimal solution We see this next 32 Strict Convexity and Uniqueness of Optimal Solutions Theorem 3 Consider an optimization problem minf(x) st x2;1 2 / 3 0 W h a t C u s e D m o c r i z n?

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$ 3 ' % 4!2 Necessary Optimality Conditions 21 Geometric Necessary Conditions AsetC ⊆ n is a cone if for every x ∈ C, αx ∈ C for any α>0 AsetC is a convex cone if C is a cone and C is a convex set Suppose x¯ ∈ SWe have the following definitions • F0= {d ∇f(¯x)tdVd bd e c o a e w x^ ^ l s k Y o o P y z M O z O P O P { / 9 3 2 9 G 0 / @ A3 9 1 / } 3 ?

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