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In fact, provided the condition above holds (ie, there exists a feasible x with ¯cTx ≥ α) we can solve the problem (1) via convex optimization We make the change of variables y = x ¯cTx−α, s = 1 ¯cTx−α, so x = y/s This yields the problem minimize q yTRy subject toT = E(X tX 0) = a2E(cos(Θ)cos(λtΘ)) = a2E 1 2 {cos(λt2Θ)cos(λt)} = a2 2 1 2π Z 2π 0 cos(λt2θ)cos(λt)dθ = a2 2 cos(λt) So ρ t = cos(λt) The spectrum is F where γ t = R π −π eitω dF(ω) Try the discrete distribution for F, F(λ) = F(−λ) = c, a constant, F(ω) = 0 otherwise Then γ t = eitλce−itλc = ccos(tλ3 Is it possible for two independent random variables X and Y (not necessarily with the same distributions) to satisfy PX < Y = 1?

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X yA y x Resultant force Over a body of constant thickness in x and y W n i F z W i 1 W dW Location x, y is the equivalent location of the force W from all W i's over all x & y locations (with respect to the moment from each force) from M x W xW n i y i i 1 W W xdW x xdW x OR W x W x M y W y W n i x i i 1 WIf X and Y are independent, then E(es(XY )) = E(esXesY) = E(esX)E(esY), and we conclude that the mgf of an independent sum is the product of the individual mgf's Sometimes to stress the particular rv X, we write M X(s) Then the above independence property canY=w(X)*f(X) c( mean(Y), var(Y) ) 1 Notice that the integral calculation is still correct, but with a variance this is approximately 1/10 of the simple monte carlo integral approximation This is one case where importance sampling provided a substantial increase in precision A plot of the integrand

Y respectively The covariance, denoted with cov(X;Y), is a measure of the association between Xand YSince x,y ∈ R we have E(ˆx− x)2 = TrΣ est = ΣxxyΣ −1 y Σ T xy, where Σx = σ2 x, Σxy = σxy, Σy = σ 2 y So we have E(ˆx− x)2 = σ2 x − σ2 xy σ2 y Of course we have E(¯x− x)2 = σ2 x, so η = E(ˆx −x)2 E(¯x −x)2 = (σ2 x − σ2 xy σ2 y)/σ2 x = 1− σxy σxσy!2 =So if X and Y are uncorrelated, then the variance of the sum is the sum of the variances Recall that independent implies uncorrelated but not vice versa Covariance is symmetric Cov(Y,X)=Cov(X,Y), and covariances of sums can be expanded as Cov(X Y,ZW)=Cov(X,Z)Cov(X,W)Cov(Y,Z)Cov(Y,W) Note that for c aconstant, Cov(X,c)=0, Cov(cX,Y

Yc(t) = c ej!t = c xc(t) where c = Aej') The efiect of a linear dynamical system on a complexvalued sinusoidal can be characterized in terms of a multiplication with a complex number;Mean of Y (% pay increase) in the population of units coded 0 on X (ie, females) 1) That is, β0 is µ 0 where µ 0 is the mean of the dependent variable for the group coded 0 2) Remember the expected value of a random variable is its mean or E(Y i) = µ ii β1 is the "effect," so to speak, of "moving" or changing fromY 2 − e x y − e x = 0 This is a second degree polynomial in y;

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2 Determining random errors 3 What is the range of possible values?Yc(t) = c xc(t) If we can write c = Aej', then A is the amplitude and ' is the phase of the output relative to the amplitude and phase of the input 32X y z p x p y p z fl fl fl fl fl fl = ¡ yp z ¡zp y ¢ ^i ¡ zp x ¡xp z ^j xp y ¡yp x ^j = L x ^iL y ^jL z ^j Let's focus on one component of angular momentum, say L x = yp z ¡ zp y On the right side of the equation are two components of position and two components of linear momentum Quantum mechanically, all four quantities

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In addition, the mean and variance of Y are E Y = 1 λ VarY = 1 λ2 (2) Since VarY = 25, we must have λ = 1/5 (b) The expected value of Y is EY = 1/λ = 5, so E Y2 = VarY (EY)2 = 50 (3) (c) P Y > 5 = Z ∞ 5 fY (y) dy = −e−y/5 ∞ 5 = e−1 (4)At Texas A&M UniversityCommerce, we educate, discover and achieve Our public university offers programs in health sciences, business, education, agriculture, humanities and the artsY=x w/2 xy=1 X 1 w w−x 1 Y x x To distinguish between the random variables and their values, I have been careful here to use capital letters for the random variable names and lower case letters for the specific values they take For values of W in the range 0 ≤ w ≤ 1, we integrate over the shaded area in the figure to the left A

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C A Bouman Digital Image Processing 2 Properties of Chromaticity Coordinates x = X X Y Z y = Y X Y Z z = Z X Y Z • xyz =1) C1 = 5⇥103 y x=5 m = 0) 50·(5)5 3 C1 ·(5)C2 = 0 C2 = 3⇥103 Hence, the equations of the elastic curve and the slope of the curve y = 1 EI 50x5 3 (5⇥103)x 3⇥103 dy dx = 1 EI 250x4 3 5⇥103 Maximum deflection at the tip y x=0 = 3⇥103 EI)ymax = 3⇥103 EI = w0L4 30EI Problem 5 Estimate the deflection curve for the beam shown Ay By w0 L L/2 x yExample 5 X and Y are jointly continuous with joint pdf f(x,y) = (e−(xy) if 0 ≤ x, 0 ≤ y 0, otherwise Let Z = X/Y Find the pdf of Z The first thing we do is draw a picture of the support set (which in this case is the first

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The DFT pair X(k) = NX−1 n=0 x(n)e−j2πkn N analysis x(n) = 1 N NX−1 k=0 X(k)ej2πkn N synthesis Alternative formulation X(k) = NX−1 n=0 x(n)Wkn ←−W = e−j2 N π x(n) = 1 N NX−1 k=0 X(k)W−kn EE 524, Fall 04, # 5 3A)w B)x C)y D)z E)x and y 19) )The normal boiling point of the substance with the phase diagram shown above is _____ °C A)10 B) C)30 D)40 E)50 ) 21)The phase diagram of a substance is shown above The area labeled _____ indicates the gas phase for the substance A)w B)x C)y D)z E)ySolution for (xy)=50 equation Simplifying (x y) = 50 Remove parenthesis around (x y) x y = 50 Solving x y = 50 Solving for variable 'x' Move all terms containing x to the left, all other terms to the right Add '1y' to each side of the equation x y 1y = 50 1y Combine like terms y 1y = 0 x 0 = 50 1y x = 50 1y Simplifying x = 50 1y

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There are 1 words with the pattern CWGY That is, nine letterSure For example, let X 2 f0;1g be equallylikely and let Y 2 f4;5g be equallylikely Even simpler, let X = 0 and Y = 1 with probability one!

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4 For some constant c > 0 the random variable X takes theC r e k P k w y G r e e n T r e L E Wheatland Rd Woodshire Dr Greenspan Ave El ev n tS W Weisenberger Dr S a n t a A n i t a B l v d E Danieldale Rd E Danieldale Rd N C a r r i e r P k w y d Town C re k D R e d B r d R C e n t e r i D r k Lake Placid Dr S W a l t o n a W a l k e r Bl v H a m il to n D r Wildlife Blvd Si l v e r C r e e k D rβ 1 is the change in the mean of Y as X

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0 nul 1 soh 2 stx 3 etx 4 eot 5 enq 6 ack 7 bel 8 bs 9 ht 10 nl 11 vt 12 np 13 cr 14 so 15 si 16 dle 17 dc1 18 dc2 19 dc3 dc4 21 nak 22 syn 23 etb 24 can4 COST FUNCTIONS 243 C2 Positively linearly homogenousin w C(y, λw)=λC(y, w),w>0 Let the cost minimizationproblem with prices w be given by C(y, w)=min x {wx x V(y)},y∈ DomV,w>0, (4)The x vector that solves this problem will be a function of y and w, and is usually denoted x(y,w)(e) the variance of Y 4 Let Y be a random variable having mean µ and suppose that E(Y −µ)4 ≤ 2 Use this information to determine a good upper bound to P(Y −µ ≥ 10) 5 Let U and V be independent random variables, each uniformly distributed on 0,1 Set X = U V and Y = U − V Determine whether or not X and Y are

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 W x Y/60 miles Example 30 mins at 50 mph W = 50 Y = 30 W x Y/60 = 50 x 30/60 = 50 x 1/2 = 25 miles What is 50 W to L Y L?A LetterEquations Brain Teaser titled '50 = W to L Y L' 50 = W to L Y L LetterEquations Letter Equations are well known phrases or facts where the key words have been replaced with the first letter of that word These are often in the form of an equation, which contain a number, an = sign and the rest of the obscured phrase or factEYi=β0 β1Xi where β 0 is the mean of when when X = 0 (assuming this is a reasonable level of X ), or more generally the Y –intercept of the regression line;

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 When installing a NEMA 1430 device, such as your dryer receptacle, the terminals should be marked W X Y and G If you were installing a NEMA 1530 device, which does not have a neutral, the terminals would be marked X Y Z and G In each case, the X YThe fact that some of the coeffi­ cients are functions of x should not slow us down Applying the quadratic formula we get y = ex ± (−ex)2 − 4 1 (−ex) 2 1 ex± √ 2 4 y = 2 Our original equation is valid only for y > 0, and √ e2x 4ex > √ e2x = e(c) E(X) = (i) P xf(x) (ii) R xf(x)dx (d) Var(X) = (i) E(X2) 2 (ii) E(Y2) 2 (e) M(t) = (i) E etX (iv) binomial (Bernoulli) (v) Poisson 3 Continuous (a) P(Y = 3) = (i) 0 (ii) 025 (iii) 050 (iv) 075 (b) P(Y 3) = F(3) = R 3 2 x 6 dx= x 2 12 i x=3 x=2 = 3 12 2 12 = 5 12 requires (i) summation (ii) integration and is a value of a (i

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If f(x,y) is convex in x for each y ∈ A, then g(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 3–16F(x 1 x 2,y 1 y 2)=(x 1 x 2 y 1 y 2,x 1 x 2) =(x 1 y 1,x 1)(x 2 y 2,x 2) (b) I must show that for each u 2 R2 there exists v 2 R2 such that T(v)=u I must solve the equation (x 1 y 1,x 1)=(x 2,y 2);Which has solutions x 1 = y 2 and y 1 = x 2 y 2 (c) The null space of a linear operator is the set of vectors mapped to the zero vector

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4' x 250' BuilttoSuit 54' x 50' B C 360 N e w Y o r k A v e C o l l i n s S t Bar di R Arli ngto Municipal Ai rpo t Arlington Commerce Center New York Avenue Arlington, TX Park Highlights Highlights Include 557,478 square feet For more information, Please contact1X !R 1Y !R 2X !W 2X !R 2Y !W 1X !W 1Y !W 2Y !C 1!C 2 A schedule is serializable if it contains the same transactions and operations as a serial schedule and the order of all con icting operations (read/writes to the same objects by di erent transactions) is also the same In the above schedule, T 1 reads X before T 2 writes XThe Fourier transform of x(t) is X(w) = x(t)e jw dt = fet/2 u(t)e dt (S911) Since u(t) = 0 for t < 0, eq (S911) can be rewritten as X(w) = e(/ 2w)t dt 2 1 j2w It is convenient to write X(o) in terms of its real and imaginary parts X(w) 2 1j2 2 j4w 1 j2w 1 j2wJ 1 4W2 2

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Expected Value and Standard Dev Expected Value of a random variable is the mean of its probability distribution If P(X=x1)=p1, P(X=x2)=p2, n P(X=xn)=pn E(X) =Covariance and correlation Let random variables X, Y with means X;Free shipping on millions of items Get the best of Shopping and Entertainment with Prime Enjoy low prices and great deals on the largest selection of everyday essentials and other products, including fashion, home, beauty, electronics, Alexa Devices, sporting goods, toys, automotive, pets, baby, books, video games, musical instruments, office supplies, and more

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T h e 50 u n it e d s tat e s , t h e d is t r ic t of c olu mb ia a n d w or ld w id e , e x c e p t f or ita ly, b r a z il, qu e b e c , r u s s ia , u k r a in e , k a z a k h s ta n , b e la r u s , c r ime a , c u b a , ir a n , s y r ia , n ort h k or e a , and s u d a n e n t ry in t h is c on t e s t c on s t it u t e s y ou r a c cIn mathematics, the exponential function is the function =, where the base e = 2718 is Euler's number and the argument x occurs as an exponentMore generally, an exponential function is a function of the form () =, where the base b is a positive real number For real numbers c and d, a function of the form () = is also an exponential function, since it can be rewritten as4 Relative and Absolute Errors 5 Propagation of Errors, Basic Rules Suppose two measured quantities x and y have uncertainties, Dx and Dy, determined by procedures described in previous sections we would report (x ± Dx), and (y ± Dy)From the measured quantities a new quantity, z, is calculated from x and y

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