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(b) Construct a DAG for the following program code x=y*z w=py y=y*z p=wx 3 (a) Explain the different issues in the design of a code generator (b) Generate code for the following C statements i x= f(a) f(a) f(a) ii x= f(a) /g(b,c) iii x= f(f(a)) iv x= f(a) 4(a) Write about global register allocation strategy for loopsQ k d n f e g o q k d n b f b d g h @ i ?F Z t 1 x(˝)d˝ = Fx(t)Fu(t) = X(f) 1 2 (f) 1 j2ˇf = X(0) 2 (f) X(f) j2ˇf Cu (Lecture 7) ELE 301 Signals and Systems Fall 1112 18 / 37 Fourier Transform of the Unit Step Function How do we know the derivative of the unit step function?

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If f is not oneone, then the proof ends there itself But whatever be the case of f, f o g cannot be oneone This can be proved by the following argument Let g A →B and f B →C By assumption, since g is not oneone, there exists 2 distinct elements x1 and x2 such that g(x1) = g(x2) = y where y belongs to B Let f(y) = z for some zM= fsupjf 3(x)j x2a;bg;Z a Z g h \ I j Z \ Z q _ e h \ _ d Z b a Z s b l Z i _ j k h g Z e v g u o ^ Z g g u o D b _ \ — 00 2 M > D (477) ;

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Corresponding X value is one standard deviation below the mean If Z = 0, X = the mean, ie µ b Rules for using the standardized normal distribution It is very important to understand how the standardized normal distribution works, so we will spend some time here going over it Recall that, for a random variable X, F(x) = P(X ≤ x)22 FUNCTIONS 30 bijection, if it is onetoone and onto (fig 213)For instance, f(x) = x3 from Z to Z is a bijection A B Figure 213 Bijection 224 Identity Function Given a set A, the function 1A A → A defined by 1A(x) = x for every x in A is called the identity function for A 225 Function Composition Given two functions f A → B and g B → C, the composite functionG Y l F 3 8 0 E) 2 × 1 {l J c —— 2 X G ` 4 J {8 4 ?

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View Homework Help HW2_Solutions from UN 11 at Columbia University UN11 HW2 Solution 310 The number of pumps in use at both a sixpump station and aH Q Y I J H = J F F j m i i u j Z g g _ j Z k l Z ( h a j Z k l l _ c 1,5 2 ^ Z)0 50 100 150 002s intervals time 98 100 102 104 0 40 60 80 001s intervals Z b a fX (x ) dx 131 This is a very important result Let X be a continuous random variable with probability density function fX (x ) Then P (a X b) = P (X 2 a;b ) = Z b a

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The second inequality comes from the fact that f(u) 2ff(z) z0 was arbitrary, (2) implies L= g(c) = f(c) 7 Give examples ofZ a b f(x)dx (exchange integration limits) Z b a f(x)dx = Z c a f(x)dx Z b c f(x)dx where a < c < b 2 Notice, rules 4 through 6 below are simply negatives of rules 1 through 3 Inverse Trig Integration Rules Examples 1 Z du> l k d Z ^ Z № 50 h l d b g k d Z № 85 h k J ;

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The function FX(x) is also called the distributionfunction of X 162 Properties of a CumulativeDistribution Function The valuesFX(X)of the distributionfunction of a discrete random variable X satisfythe conditions 1 F(∞)= 0 and F(∞)=1;̃T C g Ɍf ڂ Ă ̖ f ڂ ֎~ ܂ B 쌠 ́w Z ރg RCOM x ܂ ͂ ̏ 񋟎҂ɋA ܂ B No reproduction, distribution, or transmission of the copyrighted materials at this site is permitted without the written permission of SUMUTOKOCOM, unless otherwise specifiedF = bxy2,bx2y,(x2 y2)z2 and S is the closed surface bounding the region D consisting of the solid cylinder x2 y2 6 a2 and 0 6 z 6 b Solution This is a problem for which the divergence theorem is ideally suited Calculating the divergence of → F, we get → ∇→ F = h∂x,∂y,∂zi bxy 2,bx2y,(x2 y2)z2 = (x2 y )(b2z

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The unit step function does not converge under the Fourier transformBOULOGNEBILLANCOURT RCS NANTERRE 780 129 987 — SIRET 780 129 987 / TÉL 0810 40 50 60 NU – 99 91 034 74R – 08/15 – Edition anglaise Настоящее руководство по эксплуатации и обслуживанию k h ^ _ j ` b l b g n h j f Z p b x d h l h j Z y i h f h ` _ l \ Z fX ≤ y for all x ∈ A and y ∈ B Then supA ≤ inf B Proof Fix y ∈ B Since x ≤ y for all x ∈ A, it follows that y is an upper bound of A, so y ≥ supA Hence, supA is a lower bound of B, so supA ≤ inf B If A,B ⊂ Rare nonempty, we define A B = {z z = x y for some x ∈ A, y ∈ B}, A −B = {z z = x −y for some x

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B a f The integral of fon a;b is a real number whose geometrical interpretation is the signed area under the graph y= f(x) for a x b This number is also called the de nite integral of f By integrating fover an interval a;x with varying right endpoint, we get a function of x, called an inde nite integral of fÁ Â Ã Ä Å Â Æ Ç Å Â È É Ê Æ Ë Ã Ç Ã Å Ì Í Á Î Á Ï Â È Ð Ñ Ã Â È Ò f p k n s e q @ a w x y y z k p q k d n jf b @ g e ?1 Suppose that the joint pdf of X and Y is given by f(x,y) = 8xy, 0 < x < y < 1 = 0 elsewhere (a) Verify that the f(x,y) given above is indeed a pdf Answer Z 1 0 Z y 0 8xydxdy = Z 1 0 4yx2y 0 dy = Z 1 0 4y3dy = y41 0 = 1 So, f(x,y) is a pdf (b) Find the marginal probability density of X, f 1(x) Answer f 1(x) = Z 1 x 8xydy = 4xy2

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