Find the coefficient of the following te zy in (z + y)*

Q: (a) Show that y for the given z = e" In x+ xy

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Q: Show that for some number a > 1, we have 1- <= In(a) <= a – 1. Hint: In(a) = S"dx, please check the…

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Q: If the moment-generating function of X is M(t)= e, then the moment-generating 3t function of Y=1+-X…

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Q: A. let w-e te Find W2

A: We can solve this using u/v rule

Q: Let X ~ Poisson(λ). Derive the moment generating function of X and use that to find E(X3)

A: Solution: Let X~Poisson(λ) The probability distribution of X is  P(x)=e-λλxx!    x=0,1,2,3,..

Q: H) 9f zyzec then prove thal nl x=4=Z, (alogen) %3D

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Q: (b) State (without proof) the dimension formula for the linear transforma- tion L:V → W where both…

A: Dimension theorem

Q: (a) Let X have the following pdf : S2(x - 1) 1<z<2 S(2) = elsewhere. Compute E(X'), E(2X-1)

A: As per our guidelines we are suppose to answer first question.

Q: 3.10 Show that an arbitrary function f(x) can be expanded on the interval 0<xsa in a modified…

A: Let f(x)=∑AnJYyrnxa Now multiply both sides by x JYyrmxa , x f(x) Jvyrmxa=∑x AnJvyrmxaJvyrnxa Now…

Q: Let Zn be the size of the nth generation in a GW process with Zo = 1, E(Z¡) = µ. Show that E(Z„Zm) =…

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Q: 11

A: Condition:

Q: 7) Use Theorem 7.1.1 (Table) to find L{f(1}} for the following: S)-(e*-e*)}° a.

A: The given function is f(t)=(e-3t-e5t)2. Expand the function implies, f(t)=e-6t+e10t-2e2t The Laplace…

Q: (a) Consider two MA(2) processes, one with e, = 0: = | and another with 0=-1 and 0; = 3. O Show that…

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Q: 9.1 ) Determine for function f(x, y) = cos(x² + y?)determine the Taylor expansion up to degree 2 at…

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Q: Prove that for the gamma function [(x) = ftx-¹e-tdt that for n a positive integer then I(n) = (n-1)!

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Q: Let SINGLEDFA = { | M is a DFA and for all s e L(M),|s| = 1}. Show that SINGLEDFA is decidable.

A: Detailed explanation mentioned below.

Q: 2. Suppose the PDF of X is given by fx(x) = 5e-5x(o)(x). Find its CDF.

A: Solution

Q: (b) f(2)= (z² + 1)(z-i) at zi and z = -i. =

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Q: Find the best rank 2 approximation to A from (1) with respect to ||:||F.

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Q: Which of the following is homogenous of smallest degree? x² + y² A X x-y B C D x-y In x - In y √√x+y

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Q: 1. Use a Chain rule to find aw and ar aw , if w = x², x = sin(r²s) əs %3D

A: Given w = x2x = sin(r2s)

Q: A random process is dėji process and a >0 is a constant. Find the PSD of Y (t) in terms of the…

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Q: Bi(n, p) and using the MGF compute Derive the moment generating function of X E[X] and Var[X].

A: Given,A random variable X~Bi(n,p)P(X=x)=nxpxqn-x    ;   X=0,1,2,.....nand q=1-p

Q: The moment generating function of X is defined by M(t)=E(etx) صواب İhi

A: True.

Q: If Σ An diverges then > can diverges O only if c>0 O only if c< 0 O only if c+0 O for any real value…

A: Given, the series ∑an diverges, then we have to find the values of c for which the series ∑can…

Q: Q3: Find the Wronskian for the following solutions. Y, = 1, Y =In

A: Wronskian= 112ln1+x1-xd1dxddx12ln1+x1-x= 112ln1+x1-x01x+11-x=1x+11-x

Q: 3. For Z = x²secy,prove that Zxy is equal toZyx

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Q: If the 2n RVs A, and B, are uncorrelated with zero mean and E(A²,) = E(B²,) = %3D o, show that the…

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Q: Find the third order Taylor polynomials for the following functions. h(x) = ln(x) for x near 1.

A: Formula:

Q: Express the limit lim E (4 cos (272;) + 6)A¤; [2, 4] over as an integral. n- 00 i=1

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Q: If /has an isolated singularity at z a and /(2)- an(z-a)" is its Laurent's expansion about z a. Then…

A: We are provided that f has isolated singularity at z=a. The Laurent's expansion about…

Q: Derive the moment generating function of X ~ Bi(n, p) and using the MGF compute E[X] and Var[X].

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Q: 6. What is the coefficient of x'y" in the expansion (4x + 5y) s. Show your work here:

A: We find coefficient of x7y8 in the expansion (4x+5y)15

Q: For these more advanced pdf's, compute the constant B required for próper nörm ization: Beta (b >…

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Q: H.W/ using convolution theorem to find L-1

A: We have to find the Laplace inverse by using convolution theorem

Q: If X ~ gamma(r, 1), show that the moment generating function of is given by if t < A. Mx(t) = { ()",…

A: Given, Mxt=λλ-tr As we know that, Ex=Mx'0Ex2=Mx''0 Now, Mx't=ddtλλ-tr=rλrλ-tr+1

Q: If the moment-generating function of X is a. M(t)=(1–21)"", 1 < ½, Find the p.d.f of X hence use it…

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Q: Find the linearization of y = ev7x at x = 16. (Use symbolic notation and fractions where needed.)

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Q: Integrate the fuackion f=g regian bonndel by y the en6* y-o ver

A: The function is fx=xyThe region is bounded by y=x, y=6-x, y=0

Q: Əz 1 Q3.If Z= f(x,y)=4e* ln(y), and x = ln (u cosv), yusin v, use chain rule to find : 201 ди and az

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Q: . Let Show that, Hence show that •2T F₁(x) = 2/1/² 0 Fn(2) = eix cos ∞ N=-∞ eix cose eine de in Jn…

A: As per the question we are given the sequence of functions Fn(x) in form of a Fourier integral. And…

Q: 8. Use the Chain Rule to find az/as and az/at. z = (y-2x)³, x = s³t², y=s+t²

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Q: -B/ls//E-x] in (6.14) is distributed as at =

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Q: Find Taylor polynemial of order 3 generated by f at a a =D0

A: Given:     f(x) = e-2x      a = 0      n = 3

Q: Let x~ possion(alpha). Show that E[x(x-1)(x-2)....(x-k)]=ak+1

A: Solution

Q: 2) Find the 3 Taylor pelynomial P,(x) of fix) = Sinx at X = * ½

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Q: 5. Show that |Logz|<|In(z)|+ t, for all z # 0.

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Q: 1- Find dwi dt at the given value for w ==+2,x = cos (1), y = sin (1),z = w =-+ at = 3

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Q: find all x EIR such that 1z +zl + lz - loog | = (2+ 2015)

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.