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4 x S T CM 3003 P ri nci pl es of P R (may sub f or S T CM 3043, 4023, or 4073) 3 x F i ne art s Choose 3 credi t s f rom A RT 2503, MUS 2503, or T HE A 2503 3 x S oci al S ci Choose 3 credi t s f rom A NT H 2233;If g(x) ≥ h(x) for all x ∈ R, then Eg(X) ≥ Eh(X) 2 E(aX bY c) = aE(X)bE(Y)c for any a,b,c ∈ R 1 Let's use these definitions and rules to calculate the expectations of the following random variables if they exist Example 1 1 Bernoulli random variable 2 Binomial random variable 3 Poisson random variable5 (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), −∞




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If 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 can be concisely expressed as M XY (s) = M X(s)M Y (s), when X and Y are independentGoogle's free service instantly translates words, phrases, and web pages between English and over 100 other languagesBy g x where a and b are constants, and g x is a differentiable function of x In chapter 64, we saw that a first order equation has a oneparameter family of solutions, and that the specification of an initial condition y x0 y0 uniquely determines a solution In the case of second order equations, the basic theorem is this




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G E O G 2613;C 0 if Aoccurs P(I A =1) C= P(A) and P(I A =0) = P(A) The expectation of this indicator (noted I A) is E(I A)=1*P(A) 0*P(AC) =P(A) Onetoone correspondence between expectations and probabilities If a and b are constants, then E(aXb) = aE(X) b Proof E(aXb) = sum (ax kb) p(x k) = a sum{x kp(x k)} b sum{p(x k)} = aE(X) bProblem 1 For the function g(x) graphed here, nd the following limits or explain why they do not exist Limits from Graphs 1 For the function g(x) graphed here, find the following limits or explain why they do not exist a lim x S1 g(x) b lim x 2 g(x) c lim xS 3




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Claim 1 For Φ defined in (33), Φ satisfies ¡∆xΦ = –0 in the sense of distributions That is, for all g 2 D, ¡ Z Rn Φ(x)∆xg(x)dx = g(0)Proof Let FΦ be the distribution associated with the fundamental solution Φ That is, let FΦ D !F x y g x y and q x z p w w w w 9 ''( 3 ) 9 ''( 3 ) ''( 3 ) ''( 3 ) 2 2 2 2 f x y g x y y z f x y g x y d t x z r w w w From above equations we get r = 9t which is the required PDE 111 An equation involving atleast one partial derivatives of a function of 2 or more independent variable is called PDEYc(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;




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22 WAVES DUE TO INITIAL DISTURBANCES 4 and transmitted waves in the branches are p1(t−x/c1) and p2(t−x/c2)At the junction x= 0 we expect the continuity of pressure and fluxes, hence pi(t)pr(t)=p1(t)=p2(t) (210) pi − pr Z = p1 Z1 p2 Z2 (211) Define the reflection coefficient Rto be the amplitude ratio of reflected wave to incident wave, thenIndividuals applying for licensure for professions that require !Inside EA Throughout the month of May, the ASPIRE ERG will celebrate the API community at Electronic Arts Read More Read More Don't just get the game Get more from your game Unlock exclusive rewards, membersonly content, and a library of top titles



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Check the below NCERT MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers Pdf free download MCQ Questions for Class 12 Maths with Answers were prepared based on the latest exam pattern We have provided Continuity and Differentiability Class 12 Maths MCQs Questions with Answers to help students understand theProof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ NIn this math video lesson I solve the equation g=(xc)/x , for x This is a useful skill for students who are in Algebra and will help them to better unders



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